Unit 1 Mathematics In Management - · PDF fileQuantitative Methods 1 Unit 1 Mathematics In Management ... Time Required to Complete the unit 1. ... business use of operations research"

Quantitative Methods 1

Unit 1 Mathematics In Management

Learning Outcome

After reading this unit, you will be able to:

• Analyse major business activities using mathematics and statistics

• Identify scope and significance of business mathematics and statistics

• Explain functions

• Define and use notations and solve applications of functions

• Identify special functions

Time Required to Complete the unit

1. 1st

Reading: It will need 3 Hrs for reading a unit

2. 2nd Reading with understanding: It will need 4 Hrs for reading and understanding a

unit

3. Self Assessment: It will need 3 Hrs for reading and understanding a unit

4. Assignment: It will need 2 Hrs for completing an assignment

5. Revision and Further Reading: It is a continuous process

Content Map

1.1 Introduction

1.2 Business Mathematics and Business Statistics

1.2.1 Business Mathematics

1.2.2 Business Statistics

1.3 Scope and Importance of Mathematics in Managerial Decisions

1.4 Functions-Concept

2 Quantitative Methods

1.4.1 Definition of a Function

1.4.2 Notation

1.4.3 The Vertical Line Test

1.5 Application of Functions

1.6 Special Functions

1.6.1 Tables of Special Functions

1.6.2 Notations used in Special Functions

1.6.3 Evaluation of Special Functions

1.6.4 Kinds of Functions

1.7 Summary

1.8 Self Assessment Test

1.9 Further Reading


1.1 Introduction

Quantitative methods are research techniques that are inevitably used to table

quantitative data i.e. information dealing with numbers and anything that is measurable.

Statistics, tables and graphs are the tools used to represent the results of these methods.

They must therefore be distinctly distinguished from qualitative methods.

In most physical and biological sciences, the use of either quantitative or qualitative

methods is uncontroversial and each is used when appropriate. In the social sciences,

particularly in sociology, social anthropology and psychology, the use of one or other type of

method has become a matter of controversy and even ideology, with particular schools of

thought within each discipline favouring one type of technique and rejecting the other.

Advocates of the quantitative methods are of the view that only by using such methods can

the social sciences become truly scientific, while advocates of qualitative methods argue

that quantitative methods tend to obscure the reality of the social phenomena under study

because they underestimate or neglect the non-measurable factors, which may be of utmost

importance. The modern tendency (and in reality the majority tendency throughout the

history of social science) is to use eclectic approaches. Quantitative methods might be used

with a global qualitative frame. Qualitative methods might be used to understand the

meaning of the numbers produced by quantitative methods. Using quantitative methods, it

is possible to give a precise and testable expression to qualitative ideas. This combination of

quantitative and qualitative data gathering is often referred to as mixed-methods research.

Mathematics is an essential subject and knowledge of it enhances a person's

reasoning, problem-solving skills and in general, ability to think logically. Hence it enables an

easy grasp of most subjects, whether science and technology, medicine, the economy or

business and finance. Mathematical tools and techniques such as the Theory of Chaos are

used for mapping and forecasting market trends. Statistics and probability, which are very

important branches of mathematics, are used in everyday business and economics.

Mathematics also forms an indispensible part of accounting and many accountancy

companies prefer graduates with dual degrees with mathematics, rather than just an

accountancy qualification. Financial mathematics and business mathematics are considered

two important branches of mathematics in today's world and these are examples of the

direct application of mathematics to business and economics. Examples of applied maths

such as probability theory and management science, queuing theory, time-series analysis,

linear programming all are vital for business.

In 1967, Stafford Beer characterised the field of management science as "the

business use of operations research". However, in modern times the term management


science may also be used to refer to the separate fields of organisational studies or

corporate strategy. Like operational research itself, management science (MS) is an

interdisciplinary branch of applied mathematics devoted to optimal decision planning with

strong links with economics, business, engineering and other sciences. It uses various

scientific research-based principles, strategies and analytical methods including

mathematical modelling, statistics and numerical algorithms to improve an organisation's

ability to enact rational and meaningful management decisions by arriving at optimal or near

optimal solutions to complex decision problems. In short, management sciences help

businesses to achieve their goals using the scientific methods of operational research.

The management scientist's mandate is to use rational, systematic, science-based

techniques to inform and improve decisions of all kinds. Of course, the techniques of

management science are not restricted to business applications and may be applied to

military, medical, public administration, charitable groups, political groups or community

groups.

Management science is concerned with developing and applying models and

concepts that may prove useful in helping to elucidate management issues and solve

managerial problems, as well as designing and developing new and better models of

organisational excellence.

The application of these models within the corporate sector became known as

management science.

1.2 Business Mathematics and Business Statistics

1.2.1 BUSINESS MATHEMATICS

Business mathematics is mathematics used by commercial enterprises to record and

manage business operations. Commercial organisations use mathematics in accounting,

inventory management, marketing, sales forecasting and financial analysis. Mathematics

typically used in commerce includes elementary arithmetic, elementary algebra, statistics

and probability. Business management can be made more effective by the use of more

advanced mathematics such as calculus, matrix algebra and linear programming.

Another meaning of business mathematics, sometimes called commercial math or

consumer math, is a group of practical subjects used in commerce and everyday life.


1.2.2 BUSINESS STATISTICS

Business statistics is the science of good decision making in the face of uncertainty. It

is used in many disciplines such as financial analysis, econometrics, auditing, production and

operations including services improvement and marketing research. These sources feature

regular repetitive publication of a series of data. This makes the topic of time series

especially important for business statistics. It is also a branch of applied statistics working

mostly on data collected as a by-product of doing business or by government agencies. It

provides knowledge and skills to interpret and use statistical techniques in a variety of

business applications. A typical business statistics course is intended for business majors and

covers statistical study, descriptive statistics (collection, description, analysis and summary

of data), probability and the binomial and normal distributions, test of hypotheses and

confidence intervals, linear regression and correlation.

Fig.1.1: Graph showing BSE Sensex through the Week Sep 20 to Sep 26' 2010

Study Notes


Assessment

Differentiate between Business Mathematics and Business Statistics.

Discussion

Discuss the history of Quantative Techniques and their application in Management.

1.3 Scope and Importance of Mathematics in Managerial

Decisions

Mathematics is an integral aspect of our daily life. Many executive jobs such as those

of business consultants, computer consultants, airline pilots, company directors and a host

of others find that they require a solid understanding of basic mathematics and in some

cases require detailed knowledge of mathematics. It also plays an important role in business,

like business mathematics by commercial enterprises to record and manage business

operations.

Mathematics typically used in commerce includes, elementary arithmetic such as

fractions, decimals and percentages, elementary algebra, statistics and probability. Business

management can be made more effective in some cases by the use of more advanced

mathematics such as calculus, matrix algebra and linear programming. Commercial

organisations use mathematics in accounting, inventory management, marketing, sales

forecasting and financial analysis.

The practical applications typically include checking accounts, price discounts,

markups and markdowns, payroll calculations, simple and compound interest, consumer and

business credit and mortgages. For example, while computational formulas are covered in

most study-material on interest and mortgages, the use of prepared tables based on those

formulas is also presented and emphasised. Mathematics can provide a powerful support

for business decisions. Mathematics provides many important tools for economics and other

business fields.


Why do business consultants and directors need to know math?

Business is all about selling a product or service to make money. All transactions

within a business have to be recorded in the company accounts and quite often involve large

sums of money. So, for example, you need to be able to estimate the effect of changing

numbers in the accounts when trying to work out your expected performance for the next

year. Also, businesses rely heavily on using percentages, in particular, anyone who works as

a sales person has to be quick at mental math, approximation and in working out

percentages. The more percentage discount you give a customer when you sell them a

product, the less profit your company will make (and quite often the less you will be paid!),

so it really does pay to know your math. If you work as a sales assistant, in many stores you

need to be efficient enough to calculate the cost of goods and charge the customers as

required without using the calculator. Businesses like to know that you can cope if the

machines break down and also, they believe that you can give better customer service if you

can respond to customers who know their mathematics. Here is an example of a letter

which often appears in local newspapers as "…I bought 2 of the same item at a shop priced

at Rs3.00 and gave the young sales assistant a Rs10 note and a Re1 coin, expecting to get a

Rs 5 note as change and do my bit to help prevent the store from running out of change. To

my amazement the sales assistant insisted that I had paid too much, I tried to explain to no

avail but in the end reluctantly took back my Re1 coin and was given 4 more Re1 coins as

change". Finally, there are jobs where you can escape from using any math at all refuse

collector, building labourer, farm hand etc. However, when you invest your hard earned

cash in the bank or building society or get a loan, how do you know that you are not being

taken advantage of? You need to use math to calculate compound interest rates (to see how

much your savings can grow). You also need to use math to understand the monthly

percentages, which are added to your credit cards or bank loans or you could end up paying

Rs.10, 000 in 5 year’s time for borrowing Rs 2,000 today. This is a good reason to understand

business mathematics.

In short, we can conclude that managers need to know mathematics and statistics to

take business decisions after analysing the present scenarios and then deciding on the basis

of these results. A general example can be quoted to explain this further.

(Reference: Business Standard-Monday, 27 Sept, 2010)

It may seem like a natural progression, but it has taken more than a decade-and-a-

half for telecom operators to look at the mobile phone business in India. This is because,

earlier, the market was completely skewed towards global manufacturers like Nokia, Sony

Ericsson and Samsung. However, the recent entry and growth of national mobile brands like


Micromax, Maxx, Lava, Rage and GVL has helped many understand that the Nokias of this

world can be beaten in a price-sensitive market like India.

So, the first step from being a cell service operator to a mobile phone player was

taken by none other than the telecom giant Bharti Airtel when it announced the launch of its

own range of low-priced phones. This was launched under their subsidiary phone brand

company Beetel.

The price range of these mobile phones is between Rs 1,750 and Rs 7,000. After

Bharti, others have followed the cue. Tata Indicom recently announced a QWERTY phone,

which is a co-branded product with Alcatel. The mobile phone is bundled with Yahoo

services.

This is clearly pointing towards a trend of telecom operators looking towards the

mobile phone market for revenue growth. India does about 130- odd million in new mobile

phone sales each year and, with large subscribers now coming from the semi-urban to rural

areas, low-cost handsets seem to be the order of the day. Local players like Micromax

seemed to have cracked this aspect with their competitive price (ranging between Rs 2,000

and Rs 8,000) and an excellent bundle of features that includes social networking, among

other things.

Opportunity for Telecom Operators

As telecom penetration goes rural, telecom service giants like Bharti Airtel have the

unique advantage of a retail reach that mobile phone manufacturers are unlikely to have.

The other advantage they have is the option to bundle cheap call rates and plans along with

a mobile phone sale. They can also partner with content providers for value-added services

(VAS) products and bundle the same, targeting apps for rural India.

This is also their way to offset the impending losses that they may foresee due to the

increase tariff-based competition in the mobile services business and also the ever-falling

per-second rates.

All in all, it’s a natural progression for a telecom player to look at the Indian mobile

phones business as the market has already been expanded by existing Indian players via

their cheap pricing and feature rich phones.


Study Notes

Assessment

Why do business consultants and directors need to know math? Give examples.

Discussion

Discuss the scope and importance of mathematics in managerial decisions

1.4 Functions- Concept

The mathematical concept of a function expresses the intuitive idea that one

quantity (the argument of the function, also known as the input) completely determines

another quantity (the value or the output). A function assigns a unique value to each input

of a specified type. The argument and the value may be real numbers, but they can also be

elements from any given sets, the domain and the co-domain of the function. An example of

a function with the real numbers as both its domain and co-domain is the function f(x) = 2x,

which assigns to every real number the real number with twice its value. In this case, it is

written that f(5) = 10.


Fig. 1.2: Graph of Function

Graph of example function,

Both the domain and the range in the picture are the set of real numbers between -1

and 1.5.

In addition to elementary functions on numbers, functions include maps between

algebraic structures like groups and maps between geometric objects. In the abstract set-

theoretic approach, a function is a relation between the domain and the co-domain that

associates each element in the domain with exactly one element in the co-domain. An

example of a function with domain {A,B,C} and co-domain {1,2,3} associates A with 1, B with

2 and C with 3.

There are many ways to describe or represent functions: by a formula, by an

algorithm that computes it, by a plot or a graph. A table of values is a common way to

specify a function in statistics, physics, chemistry and other sciences. A function may also be

described through its relationship to other functions, for example, as the inverse function or

a solution of a differential equation. There are many different functions from the set of

natural numbers to itself, most of which cannot be expressed with a formula or an

algorithm.

In a setting where they have numerical outputs, functions may be added and

multiplied, yielding new functions. Collections of functions with certain properties, such as

continuous functions and differentiable functions, usually required to be closed under

certain operations are called function spaces and are studied as objects in their own right in

disciplines like real analysis and complex analysis. An important operation on functions,

which distinguishes them from numbers, is the composition of functions.


Many traditions have sprouted around the use of functions because of their wide

usage. The symbol for the input to a function is often called the independent variable or

argument and is often represented by the letter x or if the input is a particular time by the

letter t. The symbol for the output is called the dependent variable or value and is often

represented by the letter y. The function itself is most often called f and thus the notation

y = f(x) indicates that a function named f has an input named x and an output named y.

Fig. 1.3: Function ƒ

A function ƒ takes an input, x and returns an output ƒ(x). One metaphor describes

the function as a 'machine' or 'black box' that converts the input into the output.

The set of all permitted inputs to a given function is called the domain of the

function. The set of all resulting outputs is called the image or range of the function. The

image is often a subset of some larger set and is called the co-domain of a function. Thus, for

example, the function f(x) = x2 could take as its domain the set of all real numbers, as its

image, the set of all non-negative real numbers and as its co-domain the set of all real

numbers. In that case, we would describe f as a real-valued function of a real variable.

Sometimes, especially in computer science, the term 'range' refers to the co-domain rather

than the image, so care needs to be taken when using the word.

It is usual practice in mathematics to introduce functions with temporary names like

ƒ. For example, ƒ(x) = 2x+1, implies ƒ (3) = 7; when a name for the function is not needed,

the form y = 2x+1 may be used. If a function is used often, it may be given a more

permanent name, for example,

Functions need not act on numbers: the domain and co-domain of a function may be

arbitrary sets. One example of a function that acts on non-numeric inputs takes English

words as inputs and returns the first letter of the input word as output. Furthermore,

functions need not be described by any expression, rule or algorithm: indeed, in some cases


it may be impossible to define such a rule. For example, the association between inputs and

outputs in a choice function often lacks any fixed rule, although each input element is still

associated to one and only one output.

A function of two or more variables is considered in formal mathematics as having a

domain consisting of ordered pairs or triples of the argument values. For example, Sum(x,y)

= x+y operating on integers is the function- sum with a domain consisting of pairs of

integers. Sum then has a domain consisting of elements like (3, 4), a co-domain of integers

and an association between the two that can be described by a set of ordered pairs like

((3,4), 7). Evaluating Sum (3,4) then gives the value 7 associated with the pair (3,4).

A family of objects indexed by a set is equivalent to a function. For example, the

sequence 1, 1/2, 1/3, ..., 1/n, ... can be written as the ordered sequence <1/n> where n is a

natural number or as a function f(n) = 1/n from the set of natural numbers into the set of

rational numbers.

Dually, a subjective function partitions its domain into disjoint sets indexed by the

co-domain. This partition is known as the kernel of the function and the parts are called the

fibers or level sets of the function at each element of the co-domain. (A non-subjective

function divides its domain into disjoint and possibly-empty subsets).

1.4.1 DEFINITION OF A FUNCTION

A function is a rule that assigns to each element x from a set known as the 'domain' a

single element y from a set known as the 'range'. For example, the function y = x 2 + 2

assigns the value y = 3 to x = 1, y = 6 to x = 2 and y = 11 to x = 3. Using this function, we can

generate a set of ordered pairs of (x, y) including (1, 3),(2, 6) and (3, 11). We can also

represent this function graphically, as shown below.

Fig. 1.4: Graph of the function y = x 2 + 2


One precise definition of a function is that it consists of an ordered triple of sets

which may be written as (X, Y, F). X is the domain of the function, Y is the co-domain and F is

a set of ordered pairs. In each of these ordered pairs (a, b), the first element a is from the

domain, the second element b is from the co-domain and every element in the domain is

the first element in one and only one ordered pair. The set of all b is known as the image of

the function. Some authors use the term "range" to mean the image, others to mean the co-

domain.

The notation ƒ:X→Y indicates that ƒ is a function with domain X and co-domain Y.

In most practical situations, the domain and co-domain are understood from context

and only the relationship between the input and output is given. Thus

is usually written as

The graph of a function is its set of ordered pairs. Such a set can be plotted on a pair

of coordinate axes. For example, (3, 9) is the point of intersection of the lines x = 3 and y = 9.

A function is a special case of a more general mathematical concept, the relation, for

which the restriction that each element of the domain appear as the first element in one

and only one ordered pair is removed (or, in other words, the restriction that each input be

associated to exactly one output). A relation is 'single-valued' or 'functional' when for each

element of the domain set the graph contains at most one ordered pair (and possibly none)

with it as a first element. A relation is called 'left-total' or simply 'total' when for each

element of the domain, the graph contains at least one ordered pair with it as a first

element (and possibly more than one). A relation that is both left-total and single-valued is a

function.

In some parts of mathematics, including Recursion Theory and functional analysis, it

is convenient to study partial functions in which some values of the domain have no

association in the graph, i.e. single-valued relations. For example, the function f such that

f(x) = 1/x does not define a value for x = 0 and thus is only a partial function from the real

line to the real line. The term total function can be used to stress the fact that every element

of the domain does appear as the first element of an ordered pair in the graph. In other

parts of mathematics, non-single-valued relations are similarly conflated with functions:


these are called multi-valued functions, with the corresponding term single-valued function

for ordinary functions.

Some authors (especially in set theory) define a function as simply its graph f, with

the restriction that the graph should not contain two distinct ordered pairs with the same

first element. Indeed, given such a graph, one can construct a suitable triple by taking the

set of all first elements as the domain and the set of all second elements as the co-domain:

this automatically causes the function to be total and subjective. However, most authors in

advanced mathematics outside of set theory prefer the greater power of expression

afforded by defining a function as an ordered triple of sets.

Many operations in set theory- such as the power set- have the class of all sets as

their domain, therefore, although they are informally described as functions, they do not fit

the set-theoretical definition above outlined.

1.4.2 NOTATION

Formal description of a function typically involves the function's name, its domain, its

co-domain and a rule of correspondence. Thus, we frequently see a two-part notation, an

example being

Where the first part is read:

• 'ƒ is a function from N to R' (one often writes informally 'Let ƒ: X → Y' to mean 'Let ƒ be a

function from X to Y') or

• 'ƒ is a function on N into R' or

• 'ƒ is an R-valued function of an N-valued variable',

and the second part is read:

• maps to

• Here, the function named 'ƒ' has the natural numbers as domain, the real numbers as

co-domain and maps n to itself divided by π. Less, formally, this long form might be

abbreviated


Where f(n) is read as 'f as function of n' or 'f of n'. There is some loss of information:

we are no longer explicitly given the domain N and co-domain R.

It is common to omit the parentheses around the argument when there is little

chance of confusion, thus: sin x; this is known as prefix notation. Writing the function after

its argument, as in x ƒ, is known as postfix notation; for example, the factorial function is

customarily written n!, even though its generalisation, the gamma function, is written Γ(n).

Parentheses are still used to resolve ambiguities and denote precedence, though in some

formal settings the consistent use of either prefix or postfix notation eliminates the need for

any parentheses.

1.4.3 THE VERTICAL LINE TEST

In the graph, each element x is assigned a single value y. If a rule assigned more than

one value y to a single element x, that rule could not be considered a function. As you may

recall from previous calculation, we can carry out a test for this property by using the

vertical line test, where we see whether we can draw a vertical line that passes through

more than one point on the graph:

Fig. 1.5: Vertical line test on the function y = x 2 + 2

It is assumed that because any vertical line would pass through only one point, y = x 2

+ 2 must be assigning only one y value to each x value and it therefore passes the vertical

line test. Thus, y = x 2 + 2 can rightfully be considered a function.

Study Notes


Assessment

1. Explain the concept, meaning and definition of a function.

2. Explain in detail:

Discussion

Discuss, what do you understand by vertical line test.

1.5 Application of Functions

Application of functions can be cited from the following basic examples. These are

the examples of applications of functions where quantities such as area, perimeter, chord

etc are expressed as function of a variable.

Problem 1: A right triangle has one side x and a hypotenuse of 10 metres. Find the area of

the triangle as a function of x.

Solution to Problem 1:

If the sides of a right triangle are x and y, the area A of the triangle is given by

A = (1 / 2) x * y

We now need to express y in terms of x using the hypotenuse, side x and

Pythagoras's theorem

10 2 = x 2 + y 2

y = sq rt [100 - x 2]

Substitute y by its expression in the area formula to obtain

A(x) = (1 / 2) x sq rt [100 - x 2 ]

Problem 2: A rectangle has an area equal to 100 cm2 and a width x. Find the perimeter as a


function of x.


If x and y are the dimensions of the rectangle, using the formula of the area we

obtain

100 = x * y

The perimeter P is given by

P = 2(x + y)

Solve the equation 100 = x * y for y and substitute y in the formula for the perimeter

P(x) = 2(x + 100 / x)

Problem 3: Find the area of a square as a function of its perimeter x.


The area of a square of side L is given by

A = L 2

The perimeter x of a square with side L is given by

x = 4 L

Solve the above for L and substitute in the area formula A above

A(x) = (x/4) 2 = x 2 / 16

Problem 4: A right circular cylinder has a radius r and a height equal to twice r. Find the

volume of the cylinder as a function of r.


The volume V of a right circular cylinder is given by

V = (area of base of cylinder) * (height of cylinder)

= π * r 2 * (2 r)

= 2 π r 3

Problem 5: Express the length L of the chord of a circle, with given radius r = 10 cm, as a

function of the arc length s. (see figure below).



Using half the angle a, we can write

sin(a / 2) = (L / 2) / r

Substitute r by 10 and solve for L

L = 20 sin(a / 2)

The relationship between arc length s and central angle a is

s = r a = 10 a

Solve for a

a = s / 10

Substitute a by s / 10 in L = 20 sin(a / 2) to obtain

L = 20 sin ( (s / 10) / 2 )

= 20 sin ( s / 20)

Problem 6: Express the distance d = d1+ d2, in the figure below, as a function of x.


d1 is the length of the hypotenuse of a right triangle of sides x and 3, hence


d1 = sq rt [32 + x

2 ]

d2 is the length of the hypotenuse of a right triangle of sides 7 - x and 5,

Hence,

d2 = sq rt [5 2 + (7 - x) 2 ]

d = d1 + d2 is given by

d = sq rt [9 + x 2 ] + sq rt [ 25 + (7 - x) 2 ]

Study Notes

Assessment

A square has an area equal to 10,000 cm2 and its side is x. Find the perimeter as a function

of x.

Discussion

Discuss Applications of Functions.

1.6 Special Functions

Special functions are particular mathematical functions which have more or less

established names and notations due to their importance in mathematical analysis,

functional analysis, physics or other applications.


There is no general formal definition but the list of mathematical functions contains

functions which are commonly accepted as special. In particular, elementary functions are

also considered special functions..

1.6.1 TABLES OF SPECIAL FUNCTIONS

Many special functions appear as solutions of differential equations or integrals of

elementary functions. Therefore, tables of integrals usually include descriptions of special

functions and tables of special functions include most important integrals; at least, the

integral representation of special functions.

Symbolic computation engines usually recognise the majority of special functions.

Not all such systems have efficient algorithms for the evaluation, especially in the complex

plane.

1.6.2 NOTATIONS USED IN SPECIAL FUNCTIONS

In most cases, the standard notation is used for indication of a special function: the

name of function, subscripts, if any, open parenthesis, then arguments, separated with

comma and then closed parenthesis. Such a notation allows easy translation of the

expressions to algorithmic languages avoiding ambiguities. Functions with established

international notations are sin, cos, exp, erf and erfc.

Sometimes, a special function has several names. The natural logarithm can be called

as Log, log or ln, depending on the context. For example, the tangent function may be

denoted Tan, tan or tg (especially in Russian literature); arctangent may be called atan, arctg

or tan − 1

. Bessel functions may be written ; usually, , ,

refer to the same function.

Subscripts are often used to indicate arguments, typically integers. In a few cases,

the semicolon (;) or even backslash (\) is used as a separator. In this case, the translation to

algorithmic languages admits ambiguity and may lead to confusion.

Superscripts may indicate not only exponentiation but also modification of a

function. Examples include:

• usually indicates

• is typically , but never


• Usually means and not ; this one typically causes

the most confusion as it is inconsistent with the others.

1.6.3 EVALUATION OF SPECIAL FUNCTIONS

Most special functions are considered a function of a complex variable. They are

analytic; the singularities and cuts are described; the differential and integral

representations are known and the expansion to the Taylor or asymptotic series are

available. In addition, sometimes there exist relations with other special functions. A

complicated special function can be expressed in terms of simpler functions. Various

representations can be used for evaluation. The simplest way to evaluate a function is to

expand it into a Taylor series. However, such representation may converge slowly if at all. In

algorithmic languages, rational approximations are typically used, although they may behave

badly in the case of complex argument(s)..

1.6.4 KINDS OF FUNCTIONS

• Rational and polynomial

As we proceed, two types of functions to be aware of are polynomial functions and

rational functions.

1. Polynomial functions

A polynomial function is any function of the form

f (x) = a 0 + a 1 x + a 2 x 2 + ....a n-1 x n-1 + a n x n

Where a 0, a 1, a 2,...a n are constants and n is a nonnegative integer. n denotes the

'degree' of the polynomial.

Here are some common names of certain polynomial functions. A second-degree

polynomial function is a quadratic function (f (x) = ax 2 + bx + c ). A first-degree polynomial

function is a linear function (f (x) = ax + b ). Finally, a zero-degree polynomial function is a

simply a constant function (f (x) = c ).

2. Rational Functions

A rational function is a function r of the form

r(x) =


Where f (x) and g(x) are both polynomial functions. For example,

r(x) =

is a rational function. Note that we must exclude from the domain of r(x) any value of

x that would make the denominator, g(x) equal zero since this would make r(x) undefined.

Thus, x = 0 is not in the domain of the function r(x) we just defined above.

• Even and odd functions

1. Even functions

An even function, f (- x) = f (x) for all x in the domain. This sort of function is

symmetric with respect to the y–axis. In these, y axis or f(x) for any negative integer of x will

be positive.

2. Odd functions

For an odd function, f (- x) = - f (x) for all x in the domain. This sort of function is

symmetric with respect to the origin.

Odd functions, such as f (x) = x 3 , are symmetric with respect to the origin

• Composite Functions

As discussed earlier, f is a function that can take an input x and transform it into an output f

(x). Similarly, f can take the output of another function such as g(x) as its input and transform

that input into f (g(x)). When two functions are combined so that the output of one function

becomes the input for the other, the resulting combined function is called a composite

function. The notation for the composite function is f (g(x)) is (f o g)(x) .

Example:

If f (x) = 3x + 4 and g(x) = 2x - 7, then how could we find (f o g)(2)?

Solved Exercises:

Question 1: Is the graph shown below that of a function?


Solution to Question 1:

• Vertical line test: A vertcal line at x = 0 for example cuts the graph at two points. The

graph is not that of a function.

Question 2: Does the equation

y 2 + x = 1

represent a function y in terms of x?


• Solve the above equation for y

y 2= 1 - x

y = + SQRT(1 - x) or, y = - SQRT(1 - x)

• For one value of x we have two values of y and this is not a function.

Question 3: Function f is defined by

f(x) = - 2 x 2 + 6 x - 3

Find f(- 2).


• Substitute x by -2 in the formula of the function and calculate f(-2) as follows

f(-2) = - 2 (-2) 2 + 6 (-2) - 3

f(-2) = -23


Question 4: Function h is defined by

h(x) = 3 x 2 - 7 x - 5

Find h(x - 2).


• Substitute x by x - 2 in the formula of function h

h(x - 2) = 3 (x - 2) 2 - 7 (x - 2) - 5

• Expand and group like terms

h(x - 2) = 3 ( x 2 - 4 x + 4 ) - 7 x + 14 - 5

= 3 x 2 - 19 x + 7

Question 5: Functions f and g are defined by

f(x) = - 7 x - 5 and g(x) = 10 x - 12

Find (f + g)(x)


• (f + g)(x) is defined as follows

(f + g)(x) = f(x) + g(x) = (- 7 x - 5) + (10 x - 12)

• Group like terms to obtain

(f + g)(x) = 3 x - 17


f(x) = 1/x + 3x and g(x) = -1/x + 6x - 4

Find (f + g)(x) and its domain.


• (f + g)(x) is defined as follows

(f + g)(x) = f(x) + g(x)

= (1/x + 3x) + (-1/x + 6x - 4)

• Group alike terms to obtain

(f + g)(x) = 9 x - 4

• The domain of function f + g is given by the intersection of the domains of f and g


Domain of f + g is given by the interval (-infinity , 0) U (0 , + infinity)


f(x) = x 2 -2 x + 1 and g(x) = (x - 1)(x + 3)

Find (f / g)(x) and its domain.


• (f / g)(x) is defined as follows

(f / g)(x) = f(x) / g(x) = (x 2 -2 x + 1) / [ (x - 1)(x + 3) ]

• Factor the numerator of f / g and simplify

(f / g)(x) = f(x) / g(x) = (x - 1) 2 / [ (x - 1)(x + 3) ]

= (x - 1) / (x + 3) , x not equal to 1

• The domain of f / g is the intersection of the domain of f and g excluding all values of x

that make the numerator equal to zero. The domain of f / g is given by

(-infinity, -3) U (-3, 1) U (1 , + infinity)

Question 8: Find the domain of the real valued function h defined by

h(x) = SQRT ( x - 2)


• For function h to be real valued, the expression under the square root must be positive

or equal to 0. Hence the condition

x - 2 >= 0

• Solve the above inequality to obtain the domain in inequality form

x >= 2

• and interval form

[2 , + infinity)

Question 9: Find the domain of

g(x) = SQRT ( - x 2 + 9) + 1 / (x - 1)


• For a value of the variable x to be in the domain of function g given above, two

conditions must be satisfied: The expression under the square root must not be negative


- x 2 + 9 >= 0

• and the denominator of 1 / (x - 1) must not be zero

x not equal to 1

Or in interval form

(-infinity, 1) U (1, + infinity)

• The solution to the inequality - x 2 + 9 >= 0 is given by the interval

[-3, 3]

• Since x must satisfy both conditions, the domain of g is the intersection of the sets

(-infinity , 1) U (1 , + infinity) and [-3 , 3]

[-3, 1) U (1, +3]

Question 10: Find the range of

f(x) = | x - 2 | + 3


• | x - 2 | is an absolute value and is either positive or equal to zero as x takes real values,

hence

| x - 2 | >= 0

• Add 3 to both sides of the above inequality to obtain

| x - 2 | + 3 >= 3

• The expression on the left side of the above inequality is equal to f(x), hence

f(x) >= 3

• The above inequality gives the range as the interval

[3, + infinity)

Study Notes


Assessment

Function f is defined by f(x) = - 2 X 2 + 6 x - 3. Find f(- 2).

Discussion

Discuss kinds of functions.

1.7 Summary

BUSINESS MATHEMATICS

Business mathematics is mathematics used by commercial enterprises to record and

manage business operations. Commercial organisations use mathematics in accounting,

inventory management, marketing, sales forecasting and financial analysis.

BUSINESS STATISTICS

Business statistics is the science of good decision making in the face of uncertainty

and is used in many disciplines such as financial analysis, econometrics, auditing, production

and operations including services improvement and marketing research.

SCOPE AND IMPORTANCE OF MATHEMATICS IN MANAGEMENT

Mathematics is used in most aspects of daily life. Many executive jobs such as those

of business consultants, computer consultants, airline pilots, company directors and a host

of others require a solid understanding of basic mathematics and in some cases require a

detailed knowledge of mathematics.

FUNCTIONS

A function assigns a unique value to each input of a specified type. The argument and

the value may be real numbers but they can also be elements from any given sets: the

domain and the co-domain of the function.

NOTATION OF A FUNCTION

Formal description of a function typically involves the function's name, its domain, its

co-domain and a rule of correspondence. Thus, we frequently see a two-part notation.


SPECIAL FUNCTIONS

Special functions are particular mathematical functions which have more or less

established names and notations due to their importance in mathematical analysis,

functional analysis, physics or other applications.


Broad Questions

1. Evaluate f(3) given that f(x) = | x - 6 | + x 2 - 1

2. Find f(x + h) - f(x) given that f(x) = a x + b

3. Find the range of g(x) = - SQRT(- x + 2) - 6

4. Find (f o g)(x) given that f(x) = SQRT(x) and g(x) = x 2 - 2x + 1

5. How do you obtain the graph of - f(x - 2) + 5 from the graph of f(x)?

Short Notes

a. Application of mathematics in business

b. Business mathematics and business statistics

c. Vertical line test

d. Kinds of functions

e. Special functions

Answers to above Questions:

1. f(3) = 11

2. f(x + h) - f(x) = a h

3. [-2 , 1]

4. (- infinity , - 6]

5. (f o g)(x) = | x - 1 |

6. Shift the graph of f 2 units to the right then reflect it on the x axis, then shift it upward 5

units.)


1.9 Further Reading

1. Statistics for Behaviour and Social Scientists, Chadha N. K., Reliance Publishing House,

1996

2. Business Statistics, Gupta S. P. and Gupta M. P., Sultan Chand, 1997

3. Basic Statistics for Management, Kazmier L. J. and Pohl N. F., Prentice Hall Inc., 1995

4. Statistics for Management, Levin Richard I. and Rubin David S, Prentice Hall Inc, 1995

5. Linear Programming and Decision Making, Narang, A.S., 1995

6. Business Statistics by Examples, Terry Sincich, Collier MacMillan Publishers, 1990


Assignment

Exercises

1. Express the area A of a disk in terms of its circumference C.

2. The width of a rectangle is w. Express the area A of this rectangle in terms of its perimeter

P and width w.

Solutions to above exercises:

1. A = C 2 / (4 Pi)

2. A = (1/2) w (P - 2w)

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Unit 2 Sequence, Series and Matrices

Learning Outcome


• Calculate Arithmetic progression- concept, sum and product

• Illustrate Geometric progression- concept, properties, product in GP

• Create Harmonic progression- concept and sum in HP

• Interpret Matrices- definition, basic operations and applications of matrices

• Apply Markov chains- concept, examples and applications Markov


1. 1st


2. 2nd

Reading with understanding: It will need 4 Hrs for reading and understanding a

unit




Content Map

2.1 Introduction

2.2 Arithmetic Progressions

2.2.1 Sum in A.P.

2.2.2 Product in A.P.

2.3 Geometric Progressions

2.3.1 Elementary Properties of G.P.


2.3.2 Geometric Series

2.3.3 Infinite Geometric Series

2.3.4 Complex Numbers

2.3.5 Product in G.P.

2.4 Harmonic Progression

2.4.1 Harmonic Series

2.4.2 Divergence

2.4.3 Partial Sums

2.5 Managerial Application of Sequence and Series

2.6 Matrices

2.6.1 Definition of Matrices

2.6.2 Notation

2.6.3 Basic Operations

2.6.4 Matrix Multiplication

2.6.5 Application of Matrices

2.7 Markov Chains

2.7.1 Concept of Markov Chains

2.7.2 Definition of Markov Chains

2.7.3 Variations

2.7.4 Reversible Markov Chains

2.7.5 Application of Markov Chains

2.8 Summary


2.10 Further Reading


2.1 Introduction

A. SEQUENCE

A sequence is a set of numbers arranged in a definite order according to some rule. A

sequence is a function, whose domain is the set N of natural numbers.

It is defined as a succession of terms arranged in a definite order and formed

according to a definite law.

An unlimited numbers of the terms in a sequence is called an infinite sequence and

the general term of a sequence is denoted by an. A sequence is a function, whose domain is

a set of integers.

F (n) =an where n = 1, 2, 3 etc.

Sequence general term an

½, 2/3, 3/4, 4/5 n/(n+1)

½, ¼, 1/8 1/2n

1/2, -2/3, ¾ (-1) n+1 n/(n+1)

1, 3, 5, 7 (2n-1)

B. SERIES

A series is the sum of the terms of a sequence. Finite sequences and series have

defined first and last terms, whereas infinite sequences and series continue indefinitely.

In mathematics, given an infinite sequence of numbers { an }, a series is informally

the result of adding all those terms together: a1 + a2 + a3 + · · ·. These can be written more

compactly using the summation symbol ∑. An example is the famous series from Zeno's

dichotomy given below:

The terms of the series are often produced according to a certain rule, such as by a

formula or by an algorithm. As there are an infinite number of terms, this notion is often

called an infinite series. Unlike finite summations, infinite series need tools from

mathematical analysis to be fully understood and manipulated. In addition to their ubiquity

in mathematics, infinite series are also widely used in other quantitative disciplines such as

physics and computer science.


Example of series

a) 2,6,10,14,...

b) 16,8,4,2...

C. MATRICES

A matrix is defined as an ordered rectangular array of numbers. Matrices can be used

to represent systems of linear equations.

Here are a couple of examples of different types of matrices:

Symmetric Diagonal Upper Triangular

Lower Triangular Zero Identity

and a fully expanded m×n matrix A, would look like this:

... or in a more compact form:

2.2 Arithmetic Progressions

An arithmetic progression or arithmetic sequence is a sequence of numbers such that

the difference of any two successive members of the sequence is a constant. For instance,

the sequence 3, 5, 7, 9, 11, 13, … is an arithmetic progression with a common difference of


2.

If the initial term of an arithmetic progression is a1 and the common difference of

successive members is d, then the nth term of the sequence is given by:

and in general

A finite portion of an arithmetic progression is called a finite arithmetic progression

and sometimes just called an arithmetic progression.

The behaviour of the arithmetic progression depends on the common difference d. If

the common difference is:

• Positive, the members (terms) will grow towards positive infinity.

• Negative, the members (terms) will grow towards negative infinity.

Examples:

Each one of the following series form an A.P.

• 1, 3, 5, 7…

• 3, 7, 11, 15…

• 15, 12, 9…

• x, x - d, x - 2d, .....

• a, a+d, a+2d, a+3d, a+4d…

The common difference is found by subtracting any term of the series from the

immediate succeeding term.

In the above example, common difference in the first is 2, in the second it is 4, in the

third it is -3, in the fourth it is -d and in the fifth it is d.

The general form of an A.P. is as follows:

a = first term, d = common difference, then A.P. is a, a+d, a+2d, a+3d,.....

In any term, the coefficient of d is less by one than the number of terms in the series.

Thus, second term is a+d

third term is a+2d


fourth term is a+3d

tenth term is a+9d

and generally, nth

term is a + (n-1)d.

If n is the number of terms and if tn is the nth term, then

tn = a+(n-1)d.

2.2.1 SUM IN A.P.

The sum of the members of a finite arithmetic progression is called an arithmetic

series.

To find the sum of a number of terms in arithmetical progression:

Let a=first term, d=common difference, l=tn=last term, s=required sum. Then,

Writing the series in the reverse order,

Adding together the two series,

Expression i is used when the first term and the last term are given and the

expression ii is used when the first and the common difference are given. In any question

involving the five quantities a, d, l, n and s, we can determine all of them if any three are

given.

Remark

• If the same quantity is added to or subtracted from every term of an A.P, then the

resulting series will be an A.P. having the same common difference.


• If every term of an A.P. is multiplied by the same quantity, the resulting series will be in

A.P.

• If every term of a series in A.P. is divided by the same quantity, the resulting series will

be an A.P.

• If three terms are given to be in A.P., it is convenient to take them as: a-d, a, a+d.

• If four terms are given to be in A.P, it is convenient to take them as:

a-3d, a-d,a+d,a+3d

• If five terms are given to be in A.P, it is convenient to take them as:

a-2d, a-d, a, a+d, a+2d

Example 2

Express the arithmetic series in two different ways:

Adding both sides of the two equations, all terms involving d cancel:

Rearranging and remembering that an = a1 + (n − 1)d:

So, for example, the sum of the terms of the arithmetic progression given by an = 3 +

(n-1)(5) up to the 50th term is

2.2.2 PRODUCT IN A.P.

The product of the members of a finite arithmetic progression with an initial element

a1, common differences d and n elements in total is determined in a closed expression by

where denotes the rising factorial and Γ denotes the gamma function. (Note,

however, that the formula is not valid when a1 / d is a negative integer or zero.)


This is a generalisation from the fact that the product of the progression

is given by the factorial n! and that the product

for positive integers m and n is given by

Taking the example from above, the product of the terms of the arithmetic

progression given by an = 3 + (n-1)(5) up to the 50th term is

Study Notes

Assessment

1. Find the sum of the first 10 numbers from this arithmetic progression 1, 11, 21, 31.

2. Find the sum of the first 1000 odd numbers.

Discussion

Discuss sequence, series and matrices.


2.3 Geometric Progressions

A geometric progression, also known as a geometric sequence, is a sequence of

numbers where each term after the first is found by multiplying the previous one by a fixed

non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a

geometric progression with a common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric

sequence with a common ratio 1/2. The sum of the terms of a geometric progression is

known as a geometric series.

Thus, the general form of a geometric sequence is

and that of a geometric series is

where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence's start

value.

nth

term of the geometric progression is,

an=ar (n-1)

2.3.1 ELEMENTARY PROPERTIES OF G.P.

The n-th term of a geometric sequence with initial value a and common ratio r is

given by

Such a geometric sequence also follows the recursive relation

for every integer

Generally, to check whether a given sequence is geometric, one simply checks

whether all successive entries in the sequence have the same ratio.

The common ratio of a geometric series may be negative, resulting in an alternating

sequence with numbers switching from positive to negative and back. For instance,

1, −3, 9, −27, 81, −243, …

is a geometric sequence with a common ratio of −3.

The behaviour of a geometric sequence depends on the value of the common ratio.

If the common ratio is:


• Positive, the terms will all be the same sign as the initial term.

• Negative, the terms will alternate between positive and negative.

• Greater than 1, there will be exponential growth towards positive infinity.

• 1, the progression is a constant sequence.

• Between −1 and 1 but not zero, there will be exponential decay towards zero.

• −1, the progression is an alternating sequence

• Less than −1, for the absolute values there is exponential growth towards positive and

negative infinity (due to the alternating sign).

Geometric sequences (with common ratio not equal to −1,1 or 0) show exponential

growth or exponential decay, as opposed to the linear growth (or decline) of an arithmetic

progression such as 4, 15, 26, 37, 48, … (with common difference 11). This result was taken

by T.R. Malthus as the mathematical foundation of his book Principle of Population. Note

that the two kinds of progression are related: exponentiation of each term in an arithmetic

progression yields a geometric progression, while taking the logarithm of each term in a

geometric progression with a positive common ratio yields an arithmetic progression.

2.3.2 GEOMETRIC SERIES

A geometric series is the sum of the numbers in a geometric progression:

We can find a simpler formula for this sum by multiplying both sides of the above

equation by 1 − r and we will see that

since all the other terms cancel. Rearranging (for r ≠ 1) gives the convenient formula

for a geometric series:


If one were to begin the sum not from 0 but from a higher term, say m, then

Differentiating this formula with respect to r, allows us to arrive at formulae for sums

of the form

For example:

For a geometric series containing only even powers of r multiply by 1 − r2:

Then

For a series with only odd powers of r

and

2.3.3 INFINITE GEOMETRIC SERIES

An infinite geometric series is an infinite series, whose successive terms have a

common ratio. Such a series converges if and only if the absolute value of the common ratio

is less than one ( | r | < 1 ). Its value can then be computed from the finite sum formulae

Since:


Then:

For a series containing only even powers of r,

and for odd powers only,

In cases, where the sum does not start at k = 0,

The formulae given above are valid only for | r | < 1. The latter formula is valid in

every branch of algebra, as long as the norm of r is less than one and also in the field of p-

adic numbers if | r |p < 1. As in the case for a finite sum, we can differentiate to calculate

formulae for related sums. For example,

This formula only works for | r | < 1 as well. From this, it follows that, for | r | < 1,

Also, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + · · · is an elementary example of a

series that converges absolutely.

It is a geometric series, whose first term is 1/2 and whose common ratio is 1/2, so its

sum is

The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + · · · is a simple example of

an alternating series that converges absolutely.


It is a geometric series, whose first term is 1/2 and whose common ratio is −1/2, so

its sum is

2.3.4 COMPLEX NUMBERS

The summation formula for geometric series remains valid even when the common

ratio is a complex number. In this case, the condition that the absolute value of r be less

than 1 becomes that the modulus of r be less than 1. It is possible to calculate the sums of

some non-obvious geometric series. For example, consider the proposition

The proof of this comes from the fact that

which is a consequence of Euler's formula. Substituting this into the original series

gives

.

This is the difference of two geometric series and thus is a straightforward

application of the formula for infinite geometric series that completes the proof.

2.3.5 PRODUCT IN G.P.

The product of a geometric progression is the product of all the terms. If all the terms

are positive, then it can be quickly computed by taking the geometric mean of the

progression's first and last term and raising that mean to the power given by the number of

terms. (This is very similar to the formula for the sum of terms of an arithmetic sequence:

take the arithmetic mean of the first and last term and multiply it with the number of

terms.)

(if a,r > 0).


Proof:

Let the product be represented by P:

.

Now, carrying out the multiplications, we conclude that

.

Applying the sum of arithmetic series, the expression will yield

.

.

We raise both sides to the second power:

.

Consequently,

and

,

which concludes the proof.

Example for Geometric Progression:

81, 27, 9... Find the nth term formula and the value of the fifth term from the given

sequence.

Solution: The common ratio to the base r = . The nth

term formula is,

an = 81( )n−1

=> an = 81 × ( )n−1

Therefore, fifth term is,

a5 = 81 × ( )5 −1

=> 81 × ( )4


=> 81 × ( )

=> a5 = 1.

Study Notes

Assessment

Question

A piece of equipment cost a certain factory Rs. 600,000. If it depreciates in value, 15%

the first year, 13.5 % the next year, 12% the third year, and so on, what will be its value

at the end of 10 years, all percentages applying to the original cost?

(1) 2,00,000

(2) 1,05,000

(3) 4,05,000

(4) 6,50,000

[Hint: The total cost being Rs. 6,00,000/100 * 17.5 = Rs. 1,05,000.]

Discussion

Discuss difference between Arithmetic Progression and Geometric Progression.


2.4 Harmonic Progression

A harmonic progression is a progression formed by taking the reciprocals of an

arithmetic progression. In other words, it is a sequence of the form

where −1/d is not a natural number. Equivalently, a sequence is a harmonic

progression when each term is the harmonic mean of the neighbouring terms.

Examples are:

12, 6, 4, 3, 12/5, 2, …

10, 30, −30, −10, −6, −30/7, …

2.4 1 HARMONIC SERIES

In mathematics, the harmonic series is the divergent infinite series:

Its name is derived from the concept of overtones or harmonics in music. For

example, the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc of the

string's fundamental wavelength. Every term of the series after the first is the harmonic

mean of the neighbouring terms; the term harmonic mean likewise is derived from music.

The harmonic series is counterintuitive to students first encountering it because it is

a divergent series in spite of the fact that each of its terms tends to zero. Thus, an infinite

sum of numbers each of which has a value tending to zero might not be finite. The

divergence of the harmonic series is also the source of some apparent paradoxes or

counterintuitive results.

For example, one paradox is the "worm on the rubber band". Suppose that a worm

crawls along a 1 metre rubber band and after each minute, the rubber band is stretched by

an additional 1 metre. If the worm travels 1 centimetre per minute, will the worm ever reach

the end of the rubber band? The answer, counter intuitively, is "yes", for after n minutes,

the ratio of the distance travelled by the worm to the total length of the rubber band is

The series gets arbitrarily large as n becomes larger. Eventually, this ratio must


exceed 1, which implies that the worm reaches the end of the rubber band. The value of n at

which this occurs must be extremely large; however, approximately e100: a number

exceeding 1040

(a one with 40 zeros after it). Although the harmonic series diverges, it

diverges very slowly.

Another example is that given a collection of identical dominoes, it is clearly possible

to stack them at the edge of a table, so that they hang over the edge of the table. The

counterintuitive result is that one can stack them in such a way as to make the overhang

arbitrarily large, provided there are enough dominoes.

2.4.2 DIVERGENCE

The harmonic series diverges to +∞. There are several well-known proofs of this fact.

• Comparison test

One way to prove divergence is to compare the harmonic series with another

divergent series:

Each term of the harmonic series is greater than or equal to the corresponding term

of the second series and therefore, the sum of the harmonic series must be greater than the

sum of the second series. However, the sum of the second series is infinite:

It follows (by the comparison test) that the sum of the harmonic series must be

infinite as well. More precisely, the comparison above proves that

for every positive integer k. This proof, due to Nicole Oresme, is a high point of

medieval mathematics. It is still a standard proof taught in mathematics classes today.

Cauchy's condensation test is a generalisation of this argument.


• Integral test

Fig. 2.1: Harmonic series

It is also possible to prove that the harmonic series diverges by comparing the sum

with an improper integral. Specifically, consider the arrangement of rectangles shown in

figure 2.1. Each rectangle is 1 unit wide and 1 / n units high, so, the total area of the

rectangles is the sum of the harmonic series:

However, the total area under the curve y = 1 / x from 1 to infinity is given by an

improper integral:

Since this area is entirely contained within the rectangles, the total area of the

rectangles must be infinite as well. More precisely, this proves that

The generalisation of this argument is known as the integral test.

• Rate of divergence

The harmonic series diverges very slowly. For example, the sum of the first 1043

terms is less than 100. This is because the partial sums of the series have logarithmic growth.

In particular,


where γ is the Euler–Mascheroni constant and εk approaches 0 as k goes to infinity.

This result is due to Leonhard Euler.

2.4.3 PARTIAL SUMS

The nth partial sum of the diverging harmonic series,

is called the nth harmonic number.

The difference between the nth harmonic number and the natural logarithm of n

converges to the Euler-Mascheroni constant.

The difference between distinct harmonic numbers is never an integer.

No harmonic numbers are integers, except for n = 1.

Study Notes

Assessment

1. Does x=3, y=4, z=6 are in harmonic progression ? how to find that ?

2. How to find the mean of a harmonic progression ?

Discussion

Discuss how Harmonic Progression different from Geometric Progression?


2.5 Managerial Application of Sequence and Series

Sequences and series, whether arithmetic or geometric, have many applications.

To work with these application problems, one needs to have a basic understanding of

arithmetic series, arithmetic sequences, harmonic series, harmonic sequences,

geometric sequences and geometric series.

For example,

i.) A theatre has 20 seats in the first row, 24 seats in the second row, 28 seats in

the third row and so on. It has 30 rows of seats in all. How many seats are there in the

theatre?

To solve this problem, we need to ask and answer some preliminary questions.

First, what is the problem asking us to do? We need to know how many seats

are there in the auditorium, which means that we are counting things and finding a

total. Also, we need to add up all the seats in each row. Since we are adding things up,

this can be looked at as a series. Although we have formulas for series problems, we

need to know if the problem is arithmetic or geometric so that we know which formula

to use.

To find out if the problem is arithmetic or geometric, look at the pattern in the

problem. There are 20 seats in the first row, 24 in the second row and 28 in the third

row. Each row has four more seats than the one before it. Since we are adding four to

each row, this is an arithmetic sequence of numbers that we will be adding up.

Thus, we now know that our goal is to find an arithmetic series. The formula for

an arithmetic series is

To solve this problem we need n, a1 and an. In this problem, n will be equal to 30

because we are being asked to find out how many seats are there in all 30 rows or to

add up the seats in the 30 rows. The first term in the sequence, a1, is 20 because the

problem tells us that the first row has 20 seats. The only thing left to do is to find an

which will be a30.

To find a30, we need the formula for the sequence and then we substitute n = 30.

The formula for an arithmetic sequence is


We already know that is a1 = 20, n = 30 and the common difference, d, is 4. So

now we have

Thus, we now know that there are 136 seats on the 30th row. We can use this

back in our formula for the arithmetic series.

ii) You go to work for a company that pays one rupee on the first day, Rs. 2 on

the second day, Rs. 4 on the third day and so on. If the daily wage keeps doubling, what

will you total income be for working 31 days?

The problem is geometric as the problem states that the salary from the

previous day is doubled or multiplied by 2. When the same number is multiplied each

time, it is a geometric sequence. Now, the question of arises: what we need to do with

this geometric sequence?

The problem wants to know the total income after 31 days. While dealing with

total amounts, like in the previous example, we need to add the terms in a sequence. In

this case, since we will be adding terms in a geometric sequence, we will be finding a

geometric series. Thus, we need the formula for a geometric series.

We need to know n, a1 and r. We are told r = 2 when the problem says doubling

and n = 31 since that’s how many things we need to add up. We also know that the first

term is 0.01 (the decimal amount for one rupee penny). This should give us enough

information to find the answer.


More than Rs. 21.474 lakhs for 31 days work.

Practice Questions:

1. Logs are stacked in a pile with 24 logs on the bottom row and 15 on the top row.

There are 10 rows in all with each row having one more log than the one above it.

How many logs are in the stack?

2. Each hour, a grandfather clock chimes the number of times that corresponds to the

time of the day. For example, at 3:00, it will chime 3 times. How many times does

the clock chime in a day?

3. A ball is dropped from a height of 16 feet. Each time it drops, it rebounds to 80% of

the height from which it is falling. Find the total distance travelled in 15 bounces.

4. A company is offering a job with a salary of $30,000 for the first year and a 5% raise

each year after that. If that 5% raise continues every year, find the amount of

money you would earn in a 40-year career.

Study Notes


Assessment

(a) Write the recurring decimal 0·474747….. as an infinite geometric series and

hence as a fraction.

(b) In an arithmetic sequence, the fifth term is –18 and the tenth term is 12.

(i) Find the first term and the common difference.

(ii) Find the sum of the first fifteen terms of the sequence.

Ans: (a) 47/99, (b) (i) a = –42, d = 6 (ii) S15 = 0

Discussion

Discuss application of A.P., G.P. and H.P. in real life.

2.6 Matrices

In mathematics, a matrix (plural matrices or less commonly matrixes) is a rectangular

array of numbers such as:

An item in a matrix is called an entry or an element. The example has entries 1, 9, 13,

20, 55 and 4. Entries are often denoted by a variable with two subscripts, as shown above.

Matrices of the same size can be added and subtracted entry-wise and matrices of

compatible sizes can be multiplied. These operations have many of the properties of

ordinary arithmetic, except that matrix multiplication is not commutative, i.e. AB and BA are

not equal in general. Matrices consisting of only one column or row define the components

of vectors, while higher-dimensional (e.g. three-dimensional) arrays of numbers define the

components of a generalisation of a vector called a tensor. Matrices with entries in other

fields or rings are also studied.

A major branch of numerical analysis is devoted to the development of efficient

algorithms for matrix computations, a subject that is centuries old but is still an active area

of research. Matrix decomposition methods simplify computations both, theoretically and

practically. For sparse matrices, specifically tailored algorithms can provide speedups. Such

matrices arise in the finite element method.


Specific entries of a matrix are often referenced by using pairs of subscripts.

2.6.1 DEFINITION OF MATRICES

A matrix is a rectangular arrangement of numbers. For example,

An alternative notation uses large parentheses instead of box brackets:

The horizontal and vertical lines in a matrix are called rows and columns,

respectively. The numbers in the matrix are called its entries or its elements. To specify a

matrix's size, a matrix with m rows and n columns is called an m-by-n matrix or m × n matrix,

while m and n are called its dimensions. The matrix above is a 4-by-3 matrix.

A matrix with one row (a 1 × n matrix) is called a row vector and a matrix with one

column (an m × 1 matrix) is called a column vector. Any row or column of a matrix

determines a row or column vector, obtained by removing all other rows respectively

columns from the matrix. For example, the row vector for the third row of the above matrix

A is

When a row or column of a matrix is interpreted as a value, this refers to the

corresponding row or column vector. For instance, one may say that two different rows of a


matrix are equal, meaning that they determine the same row vector. In some cases, the

value of a row or column should be interpreted as a sequence of values (an element of Rn if

entries are real numbers) rather than as a matrix, for instance, when saying that the rows of

a matrix are equal to the corresponding columns of its transpose matrix.

Most of this section focuses on real and complex matrices, i.e. matrices, whose

entries are real or complex numbers.

2.6.2 NOTATION

The specifics of matrices notation varies widely, with some prevailing trends.

Matrices are usually denoted using upper-case letters, while the corresponding lower-case

letters, with two subscript indices, represent the entries. In addition to using upper-case

letters to symbolise matrices, many authors use a special typographical style, commonly

boldface upright (non-italic), to further distinguish matrices from other variables. An

alternative notation involves the use of a double-underline with the variable name, with or

without boldface style. e.g. .

The entry that lies in the i-th row and the j-th column of a matrix is typically referred

to as the i,j, (i,j) or (i,j)th entry of the matrix. For example, the (2,3) entry of the above matrix

A is 7. The (i, j)th entry of a matrix A is most commonly written as ai,j. Alternative notations

for that entry are A[i,j] or Ai,j.

Sometimes, a matrix is referred to by giving a formula for its (i,j)th entry, often with

double parenthesis around the formula for the entry. For example, if the (i,j)th entry of A

were given by aij, A would be denoted ((aij)).

An asterisk is commonly used to refer to whole rows or columns in a matrix. For

example, ai,∗ refers to the ith row of A and a∗,j refers to the jth column of A. The set of all m-

by-n matrices is denoted (m, n).

A common shorthand is

A = [ai,j]i=1,...,m; j=1,...,n or more briefly A = [ai,j]m×n

to define an m × n matrix A. Usually the entries ai,j are defined separately for all

integers 1 ≤ i ≤ m and 1 ≤ j ≤ n. They can however, sometimes be given by one formula. For

example, the 3-by-4 matrix


can alternatively be specified by A = [i − j]i=1,2,3; j=1,...,4 or simply A = ((i-j)), where the

size of the matrix is understood.

Some programming languages start the numbering of rows and columns at zero, in

which case the entries of an m-by-n matrix are indexed by 0 ≤ i ≤ m − 1 and 0 ≤ j ≤ n − 1. This

article follows the more common convention in mathematical writing where enumeration

starts from 1.

2.6.3 BASIC OPERATIONS

There are a number of operations that can be applied to modify matrices. These are

called matrix addition, scalar multiplication and transposition. These form the basic

techniques for dealing with matrices.

Table 2.1: Basic operations for matrices

Operation Definition

Addition The sum A + B of two m-by-n matrices A and B is

calculated entry wise:

(A + B)i, j = A i, j + B i, j where 1 ≤ / ≤ m and 1 ≤/≤ n.

Scalar multiplication The scalar multiplication cA of a matrix A and a

number c (also called a scalar in the parlance of

abstract algebra) is given by multiplying every

entry of A by c:

(c A) i j = c A i j

Transpose The transpose of an m-by-n matrix A is the n-by-

m matrix AT (also denoted A

tr or

tA) formed by

turning rows into columns and vice versa:

(AT) i j = A i j


Familiar properties of numbers extend to these operations of matrices: for example,

addition is commutative, i.e. the matrix sum does not depend on the order of the

summands: A + B = B + A. The transpose is compatible with addition and scalar

multiplication, as expressed by (cA)T = c(AT) and (A + B)T = AT + BT. Finally, (AT)T = A.

Row operations are ways to change matrices. There are three types of row

operations: row switching, i.e. is interchanging two rows of a matrix, row multiplication, i.e.

multiplying all entries of a row by a non-zero constant and finally row addition, which means

adding a multiple of a row to another row. These row operations are used in a number of

ways including solving linear equations and finding inverses.

2.6.4 MATRIX MULTIPLICATION

Fig. 2.2: Matrices Multiplication


Schematic depiction of the matrix product AB of two matrices A and B.

Multiplication of two matrices is defined only if the number of columns of the left

matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is

an n-by-p matrix, then their matrix product AB is the m-by-p matrix, whose entries are given

by dot-product of the corresponding row of A and the corresponding column of B:

where 1 ≤ i ≤ m and 1 ≤ j ≤ p. For example (the underlined entry 1 in the product is

calculated as the product 1 · 1 + 0 · 1 + 2 · 0 = 1):

Matrix multiplication satisfies the rules (AB)C = A(BC) (associativity) and (A+B)C =

AC+BC as well as C(A+B) = CA+CB (left and right distributivity), whenever the size of the

matrices is such that the various products are defined. The product AB may be defined

without BA being defined, namely if A and B are m-by-n and n-by-k matrices, respectively

and m ≠ k. Even if both products are defined, they need not be equal, i.e. generally one has

AB ≠ BA,

i.e. matrix multiplication is not commutative, in marked contrast to (rational, real or

complex) numbers, whose product is independent of the order of the factors. An example of

two matrices not commuting with each other is:

whereas

The identity matrix In of size n is the n-by-n matrix in which all the elements on the

main diagonal are equal to 1 and all other elements are equal to 0,

e.g.


It is called identity matrix because multiplication with it leaves a matrix unchanged:

MIn = ImM = M for any m-by-n matrix M.

Besides, the ordinary matrix multiplication just described, there exists other less

frequently used operations on matrices that can be considered to be forms of multiplication.

This includes the Hadamard product and the Kronecker product. They arise in solving matrix

equations such as the Sylvester equation.

2.6.5 APPLICATION OF MATRICES

There are numerous applications of matrices, both in mathematics and other

sciences. Some of them merely take advantage of the compact representation of a set of

numbers in a matrix. For example, in Game Theory and economics, the payoff matrix

encodes the payoff for two players, depending on which out of a given (finite) set of

alternatives the players choose. Text mining and automated thesaurus compilation makes

use of document-term matrices such as tf-idf in order to keep track of frequencies of certain

words in several documents.

Matrices are a key tool in linear algebra. One use of matrices is to represent linear

transformations, which are higher-dimensional analogs of linear functions of the form f(x) =

cx, where c is a constant; matrix multiplication corresponds to composition of linear

transformations. Matrices can also keep track of the coefficients in a system of linear

equations. For a square matrix, the determinant and inverse matrix (when it exists) govern

the behaviour of solutions to the corresponding system of linear equations and Eigen values

and eigenvectors provide insight into the geometry of the associated linear transformation.

Matrices have many applications. Physics makes use of matrices in various domains,

for example, in geometrical optics and matrix mechanics; the latter led to detailed studying

of matrices with an infinite number of rows and columns. Graph theory uses matrices to

keep track of distances between pairs of vertices in a graph. Computer graphics uses

matrices to project 3-dimensional space onto a 2-dimensional screen. Matrix calculus

generalises classical analytical notions such as derivatives of functions or exponentials to

matrices. The latter is a recurring need in solving ordinary differential equations. Serialism

and dodecaphonism are musical movements of the 20th century that use a square

mathematical matrix to determine the pattern of music intervals.

Complex numbers can be represented by particular real 2-by-2 matrices via


under which addition and multiplication of complex numbers and matrices

correspond to each other. For example, 2-by-2 rotation matrices represent the

multiplication with some complex number of absolute value 1, as illustrated above. A similar

interpretation is possible for quaternions.

Early encryption techniques like the Hill cipher also used matrices. However, due to

the linear nature of matrices, these codes are comparatively easy to break. Computer

graphics uses matrices both to represent objects and to calculate transformations of objects

using affine rotation matrices to accomplish tasks such as projecting a three-dimensional

object onto a two-dimensional screen, corresponding to a theoretical camera observation.

Matrices over a polynomial ring are important in the study of Control Theory.

Chemistry makes use of matrices in various ways, particularly since the use of the

Quantum Theory to discuss molecular bonding and spectroscopy. Examples are the overlap

matrix and the Fock matrix used in solving the Roothaan equations to obtain the molecular

orbitals of the Hartree–Fock method.

Study Notes


Assessment

Tick the correct answer from the given choices:

1. What is the size of a Matrix,

?

a) 2 X 3

b) 3 X 2

c) 3 X 4

d) 4 X 3

2. What is entry (2,3) of the matrix,

?

a) 3

b) 4

c) 12

d) 14

3. What is the matrix 7Z if

Z = ?

a)

b)

c)

d)

4. Which of the following matrices is the identity matrix I4 ?


a)

b)

c)

d)

Discussion

Discuss practical application of Matrices in daily life.

2.7 Markov Chains

A Markov chain is a random process with the property that the next state depends

only on the current state. It is a particular type of Markov process, named for Andrey

Markov, in which the process can only be in a finite or countable number of states. Markov

chains are useful as mathematical tools for statistical modelling in modern applied

mathematics, particularly in the information sciences.

2.7.1 CONCEPT OF MARKOV CHAINS

Formally, a Markov chain is a discrete random process with the Markov property.

Often, the term "Markov chain" is used to refer to a Markov process, which has a discrete

(finite or countable) state-space. Usually, a Markov chain would be defined for a discrete set

of times (i.e. a discrete-time Markov chain) although some authors use the same

terminology where "time" can take continuous values.

A "discrete-time" random process means a system, which is in a certain state at each

"step", with the state changing randomly between steps. The steps are often thought of as

time but they can equally well refer to physical distance or any other discrete measurement.

Formally, the steps are integers or natural numbers and the random process is a mapping of

these to states. The Markov property states that the conditional probability distribution for

the system at the next step (and in fact at all future steps) given its current state depends

only on the current state of the system and not additionally on the state of the system at

previous steps:


Since the system changes randomly, it is generally impossible to predict the exact

state of the system in the future. However, the statistical properties of the system's future

can be predicted. In many applications, it is these statistical properties that are important.

The changes of state of the system are called transitions and the probabilities

associated with various state-changes are called transition probabilities. The set of all states

and transition probabilities completely characterises a Markov chain. By convention, we

assume that all possible states and transitions have been included in the definition of the

processes, thus there is always a next-state and the process goes on forever.

A famous Markov chain is the so-called "drunkard's walk", a random walk on the

number line where, at each step, the position may change by +1 or −1 with equal

probability. From any position, there are two possible transitions, either to the next or

previous integer. The transition probabilities depend only on the current position and not on

the way the position was reached. For example, the transition probabilities from 5 to 4 and 5

to 6 are both 0.5 and all other transition probabilities from 5 are 0. These probabilities are

independent of whether the system was previously in 4 or 6.

Another example is the dietary habits of a creature who eats only grapes, cheese or

lettuce and whose dietary habits conform to the following (artificial) rules: it eats exactly

once a day; if it ate cheese yesterday, it will not eat today and it will eat lettuce or grapes

with equal probability; if it ate grapes yesterday, it will eat grapes today with a probability of

1/10, cheese with a probability of 4/10 and lettuce with a probability of 5/10; finally, if it ate

lettuce yesterday, it won't eat lettuce again today but will eat grapes with a probability of

4/10 or cheese with a probability of 6/10. This creature's eating habits can be modelled with

a Markov chain since its choice depends on what it ate yesterday and not additionally on

what it ate 2 or 3 (or 4, etc) days ago. One statistical property one could calculate is the

expected percentage of the time for which the creature will eat grapes over a long period.

A series of independent events—for example, a series of coin flips—does satisfy the

formal definition of a Markov chain. However, the theory is usually applied only when the

probability distribution of the next step depends non-trivially on the current state.

Many other examples of Markov chains exist.

2.7.2 DEFINITION OF MARKOV CHAINS

A Markov chain is a sequence of random variables X1, X2, X3, ... with the Markov

property, namely that, given the present state, the future and past states are independent.


Formally,

The possible values of Xi form a countable set S called the state space of the chain.

Markov chains are often described by a directed graph, where the edges are labelled

by the probabilities of going from one state to the other states.

2.7.3 VARIATIONS

• Continuous-time Markov processes have a continuous index.

• Time-homogeneous Markov chains (or stationary Markov chains) are processes where

for all n. The probability of the transition is independent of n.

• A Markov chain of order m (or a Markov chain with memory m) where m is finite, is a

process satisfying

In other words, the future state depends on the past m states. It is possible to

construct a chain (Yn) from (Xn) which has the 'classical' Markov property as follows:

Let Yn = (Xn, Xn−1, ..., Xn−m+1), the ordered m-tuple of X values. Then Yn is a Markov

chain with state space Sm and has the classical Markov property.

• An additive Markov chain of order m where m is finite, is where

for n > m.


Example

Fig. 2.3: Markov chain

A simple example is shown in figure 2.3, using a directed graph to picture the state

transitions. The states represent whether the economy is in a bull market, a bear market or

recession, during a given week. According to the figure, a bull week is followed by another

bull week 90% of the time, a bear market 7.5% of the time and a recession the other 2.5%.

From this figure, it is possible to calculate, for example, the long-term fraction of time during

which the economy is in recession or on average how long it will take to go from recession

to a bull market.

A finite state machine can be used as a representation of a Markov chain. Assuming a

sequence of independent and identically distributed input signals (for example, symbols

from a binary alphabet chosen by coin tosses), if the machine is in state y at time n, then the

probability that it moves to state x at time n + 1 depends only on the current state.

2.7.4 REVERSIBLE MARKOV CHAIN

A Markov chain is said to be reversible if there is a π such that

This condition is also known as the detailed balance condition.

Summing over i gives

Thus, for reversible Markov chains, π is always a stationary distribution.

The idea of a reversible Markov chain comes from the ability to "invert" a conditional


probability using Bayes' Rule:

Then, given the reversibility condition,

It now appears as if time has been reversed.

2.7.5 APPLICATION OF MARKOV CHAINS

Markov chains are applied in a number of ways to many different fields. Often, they

are used as a mathematical model from some random physical process; if the parameters of

the chain are known, quantitative predictions can be made. In other cases, they are used to

model a more abstract process and are the theoretical underpinning of an algorithm.

Testing: Several theorists have proposed the idea of the Markov chain statistical test

(MCST). This is a method of conjoining Markov chains to form a "Markov blanket", arranging

these chains in several recursive layers ("wafering") and producing more efficient test sets—

samples—as a replacement for exhaustive testing. MCSTs also have uses in temporal state-

based networks.

Information Sciences: Markov chains are used throughout information processing.

Claude Shannon's famous 1948 paper, A Mathematical Theory Of Communication, which in

a single step created the field of information theory, opens by introducing the concept of

entropy through Markov's modelling of the English language. Such idealised models can

capture many of the statistical regularities of systems. Even without describing the full

structure of the system perfectly, such signal models can make very effective data

compression possible through the entropy encoding technique of arithmetic coding. They

also allow effective state estimation and pattern recognition.

Markov chains are also the basis for Hidden Markov Models, which are an important

tool in such diverse fields as telephone networks (for error correction), speech recognition

and bioinformatics. The world's mobile telephone systems depend on the Viterbi algorithm

for error-correction, while hidden Markov models are extensively used in speech recognition

and also in bioinformatics, for instance, for coding region/gene prediction. Markov chains


also play an important role in reinforcement learning.

Queuing Theory: Markov chains are the basis for the analytical treatment of queues

(Queuing Theory). This makes them critical for optimising the performance of

telecommunication networks where messages must often compete for limited resources

(such as bandwidth).

Internet Applications: The Page Rank of a webpage as used by Google is defined by a

Markov chain. It is the probability to be at page i in the stationary distribution on the

following Markov chain on all (known) WebPages. If N is the number of known webpages

and a page i has ki links then it has transition probability for all pages that are

linked to and for all pages that are not linked to. The parameter α is taken to be

about 0.85.

Markov models have also been used to analyse web navigation behaviour of users. A

user's web link transition on a particular website can be modelled using first- or second-

order Markov models and can be used to make predictions regarding future navigation and

to personalise the web page for an individual user.

Statistics: Markov chain methods have also become very important for generating

sequences of random numbers to accurately reflect very complicated desired probability

distributions, via a process called Markov chain Monte Carlo (MCMC). In recent years, this

has revolutionised the practicability of Bayesian inference methods, allowing a wide range of

posterior distributions to be simulated and their parameters found numerically.

Economics and Finance: Markov chains are used in Finance and Economics to model

a variety of different phenomena, including asset prices and market crashes. The first

financial model to use a Markov chain was from Prasad et al in 1974. Another was the

regime-switching model of James D. Hamilton (1989), in which a Markov chain is used to

model switches between periods of high volatility and low volatility of asset returns. A more

recent example is the Markov Switching Multifractal Asset Pricing Model, which builds upon

the convenience of earlier regime-switching models. It uses an arbitrarily large Markov chain

to drive the level of volatility of asset returns.

Dynamic macroeconomics is greatly dependent on Markov chains. An example is

using Markov chains to exogenously model prices of equity (stock) in a general equilibrium

setting.

Social Sciences: Markov chains are generally used in describing path-dependent


arguments, where current structural configurations condition future outcomes. An example

is the commonly argued link between economic development and the rise of capitalism.

Once a country reaches a specific level of economic development, the configuration of

structural factors, such as size of the commercial bourgeoisie, the ratio of urban to rural

residence, the rate of political mobilization, etc., will generate a higher probability of

transitioning from authoritarian to capitalist.

Mathematical Biology: Markov chains also have many applications in biological

modelling, particularly population processes, which are useful in modelling processes that

are (at least) analogous to biological populations. The Leslie matrix is one such example,

though some of its entries are not probabilities (they may be greater than 1). Another

example is the modelling of a cell's shape in dividing sheets of epithelial cells. Yet another

example is the state of i in cell membranes.

Markov chains are also used in simulations of brain function such as the simulation of

the mammalian neocortex.

Study Notes

Assessment

1. A math teacher, not wanting to be predictable, decided to assign homework based on

probabilities. On the first day of class, she drew this picture on the board to tell the

students whether to expect a full assignment, a partial assignment, or no assignment

the next day.


a. Construct and label the transition matrix that corresponds to this drawing. Label it A.

b. If students have a full assignment today, what is the probability that they will have a full

assignment again tomorrow?

c. If students have no assignment today, what is the probability that they will have no

assignment again tomorrow?

d. Today is Wednesday and students have a partial assignment. What is the probability

that they will have no homework on Friday?

e. Matrix A is the transition matrix for one day. Find the transition matrix for two days (for

example, if today is Monday, what are the chances of getting each kind of assignment

on Wednesday?).

f. Find the transition matrix for three days.

g. If you have no homework this Friday, what is the is the probability that you will have no

homework next Friday (since we are only considering school days, there are only 5 days

in a week)? Give your answer accurate to two decimal places.

h. Find, to two decimal places, the matrix to which matrix A would appear to converge

after many days.

i. Explain the meaning of your solution to problem 7h.

Answers

a.

The students might arrange the rows and, therefore the columns, in a different order.


b. 0.4

c. 0.05

d. 0.18

e.

f.

g. 0.18

h.

i. If we are looking far enough into the future (a few weeks or longer), it doesn't

matter what kind of assignment we have today. We have a 49% chance of having a full

assignment, a 33% chance of having a partial assignment and an 18% chance of not having

an assignment.

Discussion

Discuss application of Markov Chains.

2.8 Summary

Sequence: A set of numbers arranged in a definite order according to some definite

rule is called a sequence.

Series: A series is the sum of the terms of a sequence.

Arithmetic Progression: An arithmetic progression or arithmetic sequence is a

sequence of numbers such that the difference of any two successive members of the

sequence is a constant.


Geometric Progression: A geometric progression, also known as a geometric

sequence, is a sequence of numbers where each term after the first is found by multiplying

the previous one by a fixed non-zero number called the common ratio.

Harmonic Progression: A harmonic progression is a progression formed by taking the

reciprocals of an arithmetic progression.

Matrix: A matrix is a rectangular arrangement of numbers.

The horizontal and vertical lines in a matrix are called rows and columns,

respectively. The numbers in the matrix are called its entries or its elements.

Markov Chains: A Markov chain is a random process with the property that the next

state depends only on the current state. It is a particular type of Markov process, named for

Andrey Markov, in which the process can only be in a finite or countable number of states.


Exercises:

1. Find the sum of the first 10 numbers from this arithmetic progression 1, 11, 21, 31...

Solution: we can use this formula S = 1/2(2a1 + d(n-1))n

S = 1/2(2.1 + 10(10-1))10 = 5(2 + 90) = 5.92 = 460

2. Find the 5th term of the G.P 64, 16, 4... (Solution: 5th term of the given G.P is 1/4)

3. Matrices, A, B and C are given by

A = , B = and C =

If D = A(2B + 3C) find d23

Choose from the options given below:

• -56

• -109

• 19

• 324


• 217

Short Notes

a. Arithmetic progression

b. Managerial applications of sequence and series

c. Matrix and its application in business

d. Markov chains and its applications

e. Sequences and series


1. Business Statistics by Examples, Terry, Sineich, Collier MacMillan Publishers, 1990

2. Basic Statistics for Management, Kazmier, L. J. and Pohl N. F., Prentice Hall Inc., 1995

3. Business Statistics, Gupta, S P and Gupta M P, New Delhi, Sultan Chand, 1997

4. Statistics for Behaviour and Social Scientists, Chadha, N. K., Reliance Publishing House,

1996




Assignment

Write down examples from day-to-day life where matrix can be used.

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Unit 3 Frequency and Probability Distribution

Learning Outcome


• Construct frequency distribution table

• Apply the table to frequency distribution-concept

• Explain probability theory and Probability distribution

• Identify Binomial and Poisson distribution

• Describe normal and exponential distribution


1. 1st



unit




Content Map

3.1 Introduction

3.2 Frequency Distribution

3.2.1 Constructing Frequency Distribution Table

3.2.2 Class, Class Limits, Class Boundaries

3.2.3 General Rules for construction of Frequency Distribution

3.2.4 Uni-variate Frequency Tables


3.2.5 Joint Frequency Distributions

3.2.6 Applications of Frequency Distribution

3.3 Introduction to Probability

3.3.1 Probability Theory

3.4 Probability Distributions

3.4.1 Definition of Probability Distribution

3.4.2 Discrete Probability Distribution

3.4.3 Continuous Probability Distribution

3.4.4 Bayes' Theorem

3.5 Binomial Distribution

3.5.1 Properties of Binomial Distribution

3.5.2 Binomial Distribution Formula

3.5.3 Examples

3.5.4 Cumulative Binomial Property

3.6 Poisson Distribution

3.6.1 Properties of Poisson Distribution

3.6.2 Formula of Poisson Distribution

3.6.3 Example Problems

3.7 Normal Distribution

3.7.1 Definition of Normal Distribution

3.7.2 Equation of Normal Distribution

3.7.3 Properties of Normal Distribution

3.7.4 Normal Variable and Normal Curve

3.7.5 The Standard Normal Distribution

3.7.6 The Z-Table

3.8 Exponential Distribution

3.8.1 Properties of Exponential Distribution


3.8.2 Occurrence and Applications

3.9 Summary




3.1 Introduction

Numerical facts or measurements obtained in the course of an enquiry into a

phenomenon that has been marked by uncertainty, constitute statistical data. Statistical

data may be already available or may have to be collected by an investigator or an agency.

Data collected for the first time by the investigator (or on his behalf) is termed primary,

while data taken from records or data already available is termed as secondary. The

Meteorological Department regularly collects data on different aspects of weather and

climate such as amount of rainfall, humidity, maximum and minimum temperature of a

certain place. This is an example of primary data. To someone using this data for a certain

investigation afterwards, the data will be secondary.

The most important method of organising and summarising statistical data is by

constructing a distribution table. In this method, classification is done according to

quantitative magnitude. The items are classified into groups of classes according to their

increasing or decreasing order of magnitude and the number of items falling into each group

is determined and indicated.

In the science of statistics, we are not concerned about the occurrence of a single

event. The statement 'Stormy coast today' is the subject matter of the level of confidence

of the one who made it. On the other hand, in the subject field of statistics, the probability

where the generalised situation is defined is very useful for it.

The probability or chance for any comment or event will be judged on the basis of all

possible cases in which it may be true or other alternate possibilities when it may be false.

The concept of probability can be elaborated by means of two approaches viz,

mathematical approach and experimental approach. The mathematical approach concerns

the classical or Priori probability, which indicates that if an event can happen in p way and

fails to happen in q way, where the chances of occurrence of p and q are same, then the

probability of happening of p will be [p / (p+q)] and that of q will be [q / (p+q)]. This is also

called Laplace’s first Principle of probability.

In the second case, the experimental approach of probability, which is also known as

statistical or empirical probability, concerns the situation where the trial is repeated for a

large number of times under identical condition. Thus, in N trials, the event E happens t

times then the probability of happening E will be equal to that of (t/N) where N ranges up to

the infinity.

As a mathematical foundation for statistics, probability theory is essential to many

human activities that involve quantitative analysis of large sets of data. Methods of


probability theory also apply to descriptions of complex systems given only partial

knowledge of their state, as in statistical mechanics. An important discovery of twentieth

century physics was the probabilistic nature of the physical phenomena at atomic scales,

described in quantum mechanics.

Most introductions to probability theory treat discrete probability distributions and

continuous probability distributions separately. The more mathematically advanced

Measure Theory-based treatment of probability covers both the discrete, the continuous,

any mix of these two and more.

3.2 Frequency Distribution

In statistics, a frequency distribution is a tabulation of the values that one or more

variables take in a sample. Each entry in the table contains the frequency or count of the

occurrences of values within a particular group or interval. In this way, the table summarises

the distribution of values in the sample.

3.2.1 CONSTRUCTING FREQUENCY DISTRIBUTION TABLE

Example 1: We consider here how a frequency distribution table is to be constructed

in the case of a discrete variable by taking a particular example.

Suppose the marks secured by 60 students of a class are as follows:

46, 57, 23,5,12 53, 38,58,26,43

36, 63,26,48,76 45, 66,74,16,86

56, 31,58,90,32 43, 36,66,46,58

36, 59,54,48,21 36, 64,58,45,76

58, 84,68,65,59 74, 48,64,58,50

46, 53,64,57,65 58, 95,56,66,44

Construct a Frequency Distribution Table

Marks obtained are divided into 10 groups or intervals as follows:

There are marks up to 10, between 11 and 20, between 11 and 30 and so on till

between 91 and 100. Represent each mark by a tally (/), for example, corresponding to the

mark 46 we put a tally (/) in the group 41 to 50: similarly, we continue putting tallies for

each mark. We continue up to four tallies and the fifth tally is put crosswise (\) so that it

becomes clear at once that the lot contains five tallies, i.e. there are five marks in that

group. A gap is left after a lot of five tallies, before starting again to mark tallies after each


lot. The number of tallies in a class or group indicates the number of marks falling under that

group. This number is known as the frequency of that group or corresponding to that class

interval. Proceeding in this way, we get the following frequency table.

Table 3.1: Frequency Distribution of Marks Secured by 60 students

Class interval Tally

Frequency (No. of

students securing marks

which fall in the class

interval)

0 to 10 / 1

11 to 20 // 2

21 to 30 //// 4

31 to 40 //// // 7

41 to 50 //// //// // 12

51 to 60 //// //// //// 15

61 to 70 //// //// / 11

71 to 80 //// 4

81 to 90 /// 3

91 and above / 1

Total 60

We shall now consider construction of a frequency distribution table of a continuous

variable.

Example 2: The heights of 50 students to the nearest centimetre are given below:

151.1, 147.2, 145.3, 153.4, 156.5, 152.1, 159.2, 153.3, 157.4, 152.5,

144.1, 151.2, 157.3, 147.4, 150.5, 157.1, 153.2, 151.3, 149.4, 147.5,

151.1, 147.2, 155.3, 156.4, 151.5, 158.1, 149.2, 147.3, 153.4, 152.5,

149.1, 151.2, 153.3, 150.4, 152.5, 154.1, 150.2, 152.3, 149.4, 151.5,


151.1, 154.2, 155.3, 152.4, 154.5, 152.1, 156.2, 155.3, 154.4, 150.5,

Construct a Frequency Distribution Table:

We have given heights in cms. In whole numbers, heights have been recorded to the

nearest centimetre. Thus a height of 144.50 or more but less than 145.5 is recorded as 145;

a height of 145.5 or more but less than 146.5 is recorded as 146 and so on. So the class 145-

146 could also be indicated by 144.5-146.5 implying the class which includes any height

greater than or equal to 144.5 but less than 145.5; the class 147-148 could be indicated by

146.5-148.5, meaning that the class which includes any height greater than or equal to 146.5

but less than 148.5. Following this convention, the classes could be represented as: 144.5-

146.5, 146.5-148.5 and so on. The above frequency distribution should thus be represented

as follows:

Table 3.2: Frequency distribution of heights of 50 students

Height Frequency (Number of students)

144.5-146.5 2

146.5-148.5 5

148.5-150.5 8

150.5-152.5 15

152.5-154.5 9

154.5-156.5 6

156.5-158.5 4

158.5-160.5 1

Total 50

3.2.2 CLASS, CLASS LIMITS, CLASS BOUNDARIES

The interval defining a class is known as a class interval. For Table 3.2, 145-146, 147-

148... are class intervals. The end numbers 145 and 146 of the class interval 145-146 are

known as class limits; the smaller number 145 is the lower class limit and larger number 146

is the upper class limit.


When we refer to the heights being recorded to the nearest centimetre and consider

a height between 144.5(greater or equal to 144.5 but less than 146.5) as falling in that class,

the class is represented as 144.5, 146.5. The end numbers are called class boundaries, while

the smaller number 144.5 is known as the lower class boundary and the larger number 146.5

as the upper class boundary. The difference between the upper and lower class boundaries

is known as the width of the class. Here, the width is 146.5 - 144.5 = 2 cm and is the same for

all the classes. The common width is denoted by c: here c = 2 cm. Note that in certain cases,

it may not be possible to have the same width for all the classes (specially the end classes).

Note also that the upper class boundary of a class coincides with the lower class

boundary of the next class; there is no ambiguity: We have clearly indicated that an

observation less than 146.5 will fall in the class 144.5 - 146.5 and an observation equal to

146.5 will fall in the class 146.5 - 148.5.

3.2.3 GENERAL RULES FOR CONSTRUCTION OF FREQUENCY DISTRIBUTION

First, find the smallest and largest observations in the data supplied and find the

range, i.e. difference between the largest and the smallest observations.

Then divide the range into a convenient number of class intervals having equal sizes.

Sometimes, one might need to consider a slightly higher value than the exact range, to get a

convenient number of class intervals of equal size. The number of class intervals taken

should not be less than six and greater than 15. The number of observations and the order

of accuracy desired can be the basis on which the number is chosen. In choosing class

intervals, care should be taken that the midpoint of the class intervals can be properly

calculated. Thirdly, find the number of observations falling in each class interval (or between

corresponding class boundaries). An ideal way to do this is by using the tally marks that we

studied earlier. Tallies are marked in lots of five or less whenever there is less in the last lot.

Example 3: The following observations give the yield of paddy in kg from 50

experimental plots in a research station:

4.4, 3.4, 4.5, 4.8, 5.1, 5.5, 4.6, 4.7, 3.6, 3.5,

4.8, 4.2, 3.4, 5.0, 4.3, 3.6, 3.5, 4.7, 5.3, 5.4,

4.6, 4.0, 5.3, 3.6, 4.3, 5.0, 3.0, 5.8, 4.8, 4.5,

3.6, 5.0, 4.0, 3.7, 4.2, 3.4, 5.3, 5.6, 4.2, 5.8,

4.6, 6.0, 6.2, 6.7, 5.0, 6.2, 6.0, 4.8, 5.6, 6.6,


Form a Frequency Distribution Table

Here, since the smallest observation is 3.0 and the largest is 6.7, the range is 6.7 - 3.0

= 3.7. Since there are 50 observations, we make each class of size 5 as follows: 2.9 - 3.4; 3.5 -

3.9 and so on. Taking the class size or width as 0.5, we can make 8 classes as 2.9 - 3.4; 3.5 -

3.9 and so on. The class limits for the class 3.0 -3.4 are 3.0 and 3.4. The class boundaries are

2.9 and 9.4 and so on. The width of a class

c = 3.4 - 2.9 = 0.5

Any observation between 2.9 and 3.4 will fall in the first class and weights are given

to the first decimal point. We now get the following frequency table:

Table 3.3: Distribution of Yield of Paddy in 50 Experimental Plots

Class interval Tally Frequency

2.9-3.4 //// 4

3.4-3.9 //// // 7

3.9-4.4 //// /// 8

4.4-4.9 //// //// / 11

4.9-5.4 //// //// 9

5.4-5.9 //// 5

5.9-6.4 //// 4

6.4-6.9 // 2

Total 50

3.2.4 UNI-VARIATE FREQUENCY TABLES

Univariate frequency distributions are often presented as lists ordered by the

quantity showing the number of times each value appears. For example, if 100 people rate a

five-point Likert scale assessing their agreement with a statement on a scale on which 1

denotes strong agreement and 5 strong disagreement, the frequency distribution of their

responses might look like:


Table 3.4: Univariate frequency table

A different tabulation scheme aggregates values into bins such that each bin

encompasses a range of values. For example, the heights of the students in a class could be

organised into the following frequency table:

Table 3.5: Frequency table

A frequency distribution shows us a summarised grouping of data which is divided

into mutually exclusive classes and the number of occurrences in a class. It is a way of

showing unorganised data, e.g. to show results of an election, income of people for a certain

region, sales of a product within a certain period, student loan amounts of graduates, etc.

Some of the graphs that can be used with frequency distributions are histograms, line

graphs, bar charts and pie charts. Frequency distributions are used for both qualitative and

quantitative data.

3.2.5 JOINT FREQUENCY DISTRIBUTIONS

Bivariate joint frequency distributions are often presented as (two-way) contingency

tables:


Table 3.6: Marginal frequency table

The total row and total column report the marginal frequencies or marginal

distribution, while the body of the table reports the joint frequencies.

3.2.6 APPLICATIONS OF FREQUENCY DISTRIBUTION

Managing and operating frequency tabulated data is much simpler than operating on

raw data. There are simple algorithms to calculate median, mean, standard deviation etc

from these tables.

Statistical hypothesis testing is founded on the assessment of differences and

similarities between frequency distributions. This assessment involves measures of central

tendency or averages such as the mean and median and measures of variability or statistical

dispersion such as the standard deviation or variance.

A frequency distribution is said to be skewed when its mean and median are

different. The kurtosis of a frequency distribution is the concentration of scores at the mean

or how peaked the distribution appears if depicted graphically—for example, in a histogram.

If the distribution is more peaked than the normal distribution it is said to be leptokurtic; if

less peaked it is said to be platykurtic.

Letter frequency distributions are also used in frequency analysis to crack codes and

refer to the relative frequency of letters in different languages.

Study Notes


Assessment

In the US Open Tennis 2002, Max Mirnyi played 5 matches, Andy Roddick played 5

matches, Kenneth Carlsen played 2 matches, Andre Agassi played 7 matches and Pete

Sampras played 6 matches. Pick an appropriate frequency table for the data.

Answer: (a) Table 1, (b) Table 2 (c) Table 3 (4) Table 4

Discussion

Discuss application of frequency distribution in managerial decision making.

3.3 Introduction to Probability

In day-to-day life, we came across some words viz., probability, chance odds or

likelihood which indicate our uncertainty towards the phenomenon. The sample of quotes

which we find in our life that there is a chance the UK may lead the Commonwealth Games

medal tally at India, there is a likelihood of a storm at the sea coast. We use these terms in

our talk but its presentation in a mathematical form is difficult.


In other words, we can explain the concept as we consider the few simple questions

viz., “Will it rain this week or not?”, “Will a bus reach the destination in time or not?” or

“Will the coins lifted be of the same value?” etc, in all of which there is uncertainty or a

prevalence of doubt. The strength of the doubt differs as per the case or situation viz., the

tossed coin will land either showing head or tail but the probability of a baby born in the

year 2008 going on to become Prime Minister is very uncertain. This strength of doubt is

called the degree of doubt of that event.

Simply put, the probability is the ratio of the number of favourable cases to that of

the total number of equally likely or possible cases.

Hence, probability can be measured as

No. of favourable cases

Probability = ________________________________

Total number of all possible cases

3.3.1 PROBABILITY THEORY

Probability theory is the branch of mathematics concerned with the analysis of

random phenomena. The central objects of the Probability Theory are random variables,

stochastic processes and events: mathematical abstractions of non-deterministic events or

measured quantities that may either be single occurrences or evolve over time in an

apparently random fashion. Although an individual coin toss or the roll of a die is a random

event, if repeated many times, the sequence of random events will exhibit certain statistical

patterns, which can be studied and predicted. Two representative mathematical results

describing such patterns are the Law of Large Numbers and the central limit theorem.

The measure-theoretic treatment unifies the discrete and the continuous cases and

makes the difference a result of the measure that is used. Furthermore, it covers

distributions that are neither discrete nor continuous nor mixtures of the two.

An example of such distributions could be a mix of discrete and continuous

distributions. For example, a random variable, which is 0 with probability 1/2 and takes a

random value from a normal distribution with probability 1/2. It can still be studied to some

extent by considering it to have a pdf of , where δ[x] is the Dirac delta

function.

Other distributions may not even be a mix. For example, the Cantor distribution has

no positive probability for any single point and neither does it have a density. The modern

approach to Probability Theory solves these problems using Measure Theory to define the


probability space:

Given any set , (also called sample space) and a σ-algebra on it, a measure

defined on is called a probability measure if

If is the Borel σ-algebra on the set of real numbers, then there is a unique

probability measure on for any cdf and vice versa. The measure corresponding to a cdf is

said to be induced by the cdf. This measure coincides with the pmf for discrete variables and

pdf for continuous variables, thus making the measure-theoretic approach free of fallacies.

The probability of a set in the σ-algebra is defined as

where the integration is with respect to the measure induced by

Along with providing better understanding and unification of discrete and continuous

probabilities, the measure-theoretic treatment also allows us to work on probabilities

outside , as in the Theory of Stochastic Processes. For example, in order to study

Brownian motion, probability is defined on a space of functions.

Study Notes

Assessment

What do you understand by probability? Give general examples in this context.


Discussion

Discuss and find the answers of the following questions:

• A die is rolled, find the probability that an even number is obtained.

• Two coins are tossed, find the probability that two heads are obtained.

• A card is drawn at random from a deck of cards. Find the probability of getting the 3 of

diamond.

• A jar contains 3 red marbles, 7 green marbles and 10 white marbles. If a marble is

drawn from the jar at random, what is the probability that this marble is white?

3.4 Probability Distributions

In Probability Theory and statistics, a probability distribution identifies either the

probability of each value of a random variable (when the variable is discrete) or the

probability of the value falling within a particular interval (when the variable is continuous).

The probability distribution describes the range of possible values that a random variable

can attain and the probability that the value of the random variable is within any

(measurable) subset of that range.

Fig. 3.1: Normal Distribution Curve

Normal distribution is often called the "bell curve".

When the random variable takes values in the set of real numbers, the probability

distribution is described completely by the cumulative distribution function, whose value at

each real x is the probability that the random variable is smaller than or equal to x.

The concept of the probability distribution and the random variables, which they


describe underlies the mathematical discipline of Probability Theory and the science of

statistics. There is spread or variability in almost any value that can be measured in a

population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.);

almost all measurements are made with some intrinsic error; in physics many processes are

described probabilistically, from the kinetic properties of gases to the quantum mechanical

description of fundamental particles. For these and many other reasons, simple numbers are

often inadequate for describing a quantity, while probability distributions are often more

appropriate.

There are various probability distributions that show up in various different

applications. One of the more important ones is the normal distribution, which is also known

as the Gaussian distribution or the bell curve and approximates many different naturally

occurring distributions. The toss of a fair coin yields another familiar distribution, where the

possible values are heads or tails, each with a probability of 1/2.

3.4.1 DEFINITION OF PROBABILITY DISTRIBUTION

In the measure-theoretic formalisation of Probability Theory, a random variable is

defined as a measurable function X from a probability space to measurable space

. A probability distribution is the push forward measure X*P = PX −1 on .

3.4.2 DISCRETE PROBABILITY DISTRIBUTION

A probability distribution is called discrete if its cumulative distribute on function

increases only in jumps. More precisely, a probability distribution is discrete if there is a

finite or countable set, whose probability is 1.

For many familiar discrete distributions, the set of possible values is topologically

discrete in the sense that all its points are isolated points. Also, there are discrete

distributions for which this countable set is dense on the real line.

Discrete distributions are characterized by a probability mass function, p such that

3.4.3 CONTINUOUS PROBABILITY DISTRIBUTION

By one convention, a probability distribution is called continuous if its cumulative

distribution function is continuous and therefore, the probability

measure of singletons for all .

Another convention reserves the term continuous probability distribution for

absolutely continuous distributions. These distributions can be characterised by a probability


density function: a non-negative Lebesgue integrable function defined on the real numbers

such that

Discrete distributions and some continuous distributions (like the Cantor

distribution) do not admit such a density.

3.4.4 BAYES' THEOREM

In probability theory and applications, Bayes' theorem shows the relation between

two conditional probabilities which are the reverse of each other. This theorem is named for

Thomas Bayes and often called Bayes' law or Bayes' rule. Bayes' theorem expresses the

conditional probability, or "posterior probability", of a hypothesis H (i.e. its probability after

evidence E is observed) in terms of the "prior probability" of H, the prior probability of E, and

the conditional probability of E given H. It implies that evidence has a stronger confirming

effect if it was more unlikely before being observed. Bayes' theorem is valid in all common

interpretations of probability, and it is commonly applied in science and engineering.

However, there is disagreement among statisticians regarding its proper implementation.

Thomas Bayes addressed both the case of discrete probability distributions of data

and the more complicated case of continuous probability distributions. In the discrete case,

Bayes' theorem relates the conditional and marginal probabilities of events A and B,

provided that the probability of B does not equal zero:

Each term in Bayes' theorem has a conventional name:

• P(A) is the prior probability or marginal probability of A. It is "prior" in the sense that it

does not take into account any information about B.

• P(A|B) is the conditional probability of A, given B. It is also called the posterior

probability because it is derived from or depends upon the specified value of B.

• P(B|A) is the conditional probability of B given A. It is also called the likelihood.

• P(B) is the prior or marginal probability of B, and acts as a normalizing constant.

Bayes' theorem in this form gives a mathematical representation of how the

conditional probability of event A given B is related to the converse conditional probability

of B given A.


Bayes' theorem with continuous prior and posterior distributions

Suppose a continuous probability distribution with probability density function ƒΘ is

assigned to an uncertain quantity Θ. (In the conventional language of mathematical

probability theory Θ would be a "random variable") The probability that the event B will be

the outcome of an experiment depends on Θ; it is P(B | Θ). As a function of Θ this is the

likelihood function:

Then the posterior probability distribution of Θ, i.e. the conditional probability

distribution of Θ given the observed data B, has probability density function

Where the "constant" is a normalizing constant so chosen as to make the integral of

the function equal to 1, so that it is indeed a probability density function. This is the form of

Bayes' theorem actually considered by Thomas Bayes.

In other words, Bayes' theorem says:

To get the posterior probability distribution, multiply the prior probability

distribution by the likelihood function and then normalize.

More generally still, the new data B may be the value of an observed continuously

distributed random variable X. The probability that it has any particular value is therefore 0.

In such a case, the likelihood function is the value of a probability density function of X given

Θ, rather than a probability of B given Θ:

Bayes' Theorem derived via conditional probabilities.

To derive Bayes' theorem, start from the definition of conditional probability. The

probability of the event A given the event B is

Equivalently, the probability of the event B given the event A is

Rearranging and combining these two equations, we find


This lemma is sometimes called the product rule for probabilities. Discarding the

middle term and dividing both sides by P(B), provided that neither P(B) nor P(A) is 0, we

obtain Bayes' theorem:

Of course, this lemma is symmetric in A and B, since A and B are arbitrarily-chosen

symbols, and dividing by P(A), provided that it is non-zero, gives a statement of Bayes'

theorem, in which the two symbols have changed places.

Theoretical probability distribution is classified into the following:

• Binomial Distribution

• Poisson Distribution

• Normal Distribution

• Exponential Distribution

Study Notes

Assessment

1. Differentiate between Discrete Probability distribution and continous probability

distribution

2. Explain Bayes Theorem


Discussion

Discuss and solve: The table shows the probability distribution for the random variable x,

where x represents the number of CDs a person rents from a video store during a single

visit.

X p(x)

0 0.06

1 0.58

2 0.22

3 0.10

4 0.03

5 0.01

Determine whether the following is a valid probability distribution for the random variable

x.

[Hint: Since ΣP(x) = 0.97 is not equal to 1.

x = (0) (.06) + (1) (.58) + (2) (.220 + (4) (.03) + (5) (0.01)

= 1.49 CDs]

3.5 Binomial Distribution

In Probability Theory and statistics, the binomial distribution is the discrete

probability distribution of the number of successes in a sequence of n independent yes/no

experiments, each of which yields success with probability p. Such a success/failure

experiment is also called a Bernoulli experiment or Bernoulli trial. In fact, when n = 1, the

binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the

popular binomial test of statistical significance.

It is frequently used to model the number of successes in a sample of size n from a

population of size N. Since the samples are not independent (this is sampling without

replacement), the resulting distribution is a hyper geometric distribution and not a binomial

one. However, for N much larger than n, the binomial distribution is a good approximation

and is also widely used.


3.5.1 PROPERTIES OF BINOMIAL DISTRIBUTION

• The experiment has n repeated trials.

• Each trial can have two possible outcomes. One is success and the other is failure.

• Here the trials are independent.

• Mean = n * P.

• Variance = n * P * (1 – P).

• Standard Deviation = sqrt[ n * P * ( 1 – P ) ].

3.5.2 BINOMIAL DISTRIBUTION FORMULA

b(x; n, P) = nCx * Px * (1 - P)n – x

Here the Notation are,

B(x; n, P) = binomial probability

X = successes

N = number of trials

P = probability of success

nCx = number of combinations of n trials, x is success.

3.5.3 EXAMPLES

An elementary example is this: roll a standard die ten times and count the number of

fours. The distribution of this random number is a binomial distribution with n = 10 and

p = 1/6.

For another example, flip a coin three times and count the number of heads. The

distribution of this random number is a binomial distribution with n = 3 and p = 1/2.

Binomial distribution is a statistical experiment which means the number of

successes in n repeated trials of a binomial experiment. It is also called Bernoulli distribution

or Bernoulli trial.

For example,

For a clinical trial, a patient may live or die. Here the researcher faces the number of

survivors and not how much time the patient lives after treatment.

We take a coin and flip it twice. Here we calculate the count of number of

heads(successes). Thus, the binomial distribution is


Number of heads Probability

No head 0.25

One head 0.5

Two head 0.25

Example Problem(the Binomial Distribution)

A die is tossed 6 times. What is the probability of rolling fours on two occasions?

Solution

Here n = 6, x = 2, probability of success on a single trial = 1/ 6 or 01.167.

Therefore, The binomial probability is,

b( 2; 6, 0.167 ) = 6C2 * ( 0.167 )2 * ( 1 – 0.167)6 – 2

= ( 6! / 2! * (6-2)!) * 0.0279 * ( 0.833)4

= (6! / 2! * 4!) * 0.0279 * 0.481

= 15 * 0.0279 * 0.481

b( 2; 6, 0.167 ) = 0.201. Answer.

3.5.4 CUMULATIVE BINOMIAL PROBABILITY

It refers to the binomial probability which falls within a specified range that is

greater than or equal to a mentioned lower limit and less than or equal to a mentioned

upper limit.

For example,

Cumulative binomial probability of obtaining 5 or fewer heads in 10 tosses of a coin.

b( x <= 5; 10, 0.5)= b( x = 0; 10, 0.5) + b( x = 1; 10, 0.5) +…… + b ( x = 5; 10, 0.5)

Definition:

The binomial distribution is one of the distinct probability distributions. It is used

when there are exactly two equally exclusive outcomes of a trial. These outcomes are

appropriately labelled success and failure. The binomial distribution is used to find the

probability of observing ‘r’ successes in ‘n’ trials, with the probability of success on a single

trial denoted by ‘p’.

Formula:

P(X = r) = nCr p r (1-p) n-r


where,

n = Number of events.

r = Number of successful events.

p = Probability of success on a single trial.

nCr =

1-p = Probability of failure

Example of Binomial Distribution

1.Toss a coin for 12 times. What is the probability of getting exactly 6 heads?

Solution:

Step 1: Here,

Number of trials n = 12

Number of success r = 6 (since we define getting a head as success)

Probability of success on any single trial p = 0.5

Step 2: To calculate nCr formula is used.

nCr =

=

=

=

=

= 924

Step 3: Find pr.

pr = 0.5

6

= 0.015625

Step 4: To Find (1-p)n-r Calculate 1-p and n-r.

1-p = 1-0.5 = 0.5

n-r = 12-6 = 6

Step 5: Find (1-p)n-r.


= 0.56 = 0.015625

Step 6: Solve P(X = r) = nCr p r (1-p)n-r

= 924 × 0.015625 × 0.015625

= 0.2255859375

The probability of getting exactly 6 heads is 0.23

Second Example on Binomial Distribution

Suppose a die is tossed 5 times. What is the probability of getting fours twice?

Solution:

Step 1:Number of trials n = 5

Number of success r = 2

Probability of success on any single trial p = 1/6 or 0.167

Step 2:To calculate nCr formula is used.

nCr =

=

=

=

=

= 10

Step 3:Find pr.

pr = 0.1672

= 0.027889

Step 4: To Find (1-p)n-r Calculate 1-p and n-r.

1-p = 1-0.167 = 0.833

n-r = 5-2 = 3

Step 5:Find (1-p)n-r

.

= 0.8333 =0.578

Step 6:Solve P(X = r) = nCr p r (1-p)n-r

= 10 × 0.027889 × 0.578

= 0.16120

The probability of getting exactly 2 fours is0.16


Study Notes

Assessment

1. What did you understand by Binomial Distribution?

2. Explain Cumulative Binomial Distribution.

Discussion

Discuss and solve: A die is tossed 6 times. What is the Probability of getting exactly 2 fours?

[ Hint:Here n = 6, x = 2, probability of success on a single trial = 1/ 6 = 0.167.

Then p = 0.167,

p + q =1

p = 1-q

Formula P(b) = ncr pr q(n-r) or q =(1-p).

Ans. b( 2; 6, 0.167 ) = 0.201]

3.6 Poisson Distribution

In Probability Theory and statistics, the Poisson distribution is a discrete probability

distribution that expresses the probability of a number of events occurring in a fixed period

of time if these events occur with a known average rate and independently of the time since

the last event. (The Poisson distribution can also be used for the number of events in other


specified intervals such as distance, area or volume.)

The distribution was first introduced by Siméon-Denis Poisson (1781–1840) and

published, together with his probability theory, in 1838 in his work Recherches sur la

probabilité des jugements en matière criminelle et en matière civile (Research on the

Probability of Judgments in Criminal and Civil Matters). The work focused on certain random

variables N that count, among other things, the number of discrete occurrences (sometimes

called “arrivals”) that take place during a time-interval of given length.

If the expected number of occurrences in this interval is λ, then the probability that

there are exactly k occurrences (k being a non-negative integer, k = 0, 1, 2, ...) is equal to

where

• e is the base of the natural logarithm (e = 2.71828...)

• k is the number of occurrences of an event - the probability of which is given by the

function

• k! is the factorial of k

• λ is a positive real number, equal to the expected number of occurrences that occur

during the given interval. For instance, if the events occur on average 4 times per minute

and you are interested in probability for k times of events occurring in a 10 minute

interval, you would use a Poisson distribution with λ = 10×4 = 40 as your model.

As a function of k, this is the probability mass function. The Poisson distribution can

be derived as a limiting case of the binomial distribution.

• The Poisson distribution can be applied to systems with a large number of possible

events, each of which is rare. A classic example is the nuclear decay of atoms.

3.6.1 PROPERTIES OF POISSON DISTRIBUTION

• The expected value of a Poisson-distributed random variable is equal to λ. It is the same

for its variance. The higher moments of the Poisson distribution are Touchard

polynomials in λ, whose coefficients have a combinatorial meaning. In fact, when the

expected value of the Poisson distribution is 1, then Dobinski's formula says that the nth

moment equals the number of partitions of a set of size n.


• The mode of a Poisson-distributed random variable with non-integer λ is equal to ,

which is the largest integer less than or equal to λ. This is also written as floor(λ). When λ

is a positive integer, the modes are λ and λ − 1.

• Sums of Poisson-distributed random variables:

If follow a Poisson distribution with parameter and Xi are

independent, then

also follows a Poisson distribution, whose parameter is the sum of the component

parameters. A converse is Raikov's Theorem, which says that if the sum of two independent

random variables is Poisson-distributed, then so is each of those two independent random

variables.

• The sum of normalised square deviations is approximately distributed as chi-square if

the mean is of a moderate size (λ > 5 is suggested).

If are observations from independent Poisson distributions with means

then

• The moment-generating function of the Poisson distribution with expected value λ is

• All of the cumulants of the Poisson distribution are equal to the expected value λ. The

nth factorial moment of the Poisson distribution is λn.

• The Poisson distributions are infinitely divisible probability distributions.

• The directed Kullback-Leibler divergence between Pois(λ) and Pois(λ0) is given by

3.6.2 FORMULA OF POISSON DISTRIBUTION

Formula for Poisson distribution is

• e is function of log e=2.718


• is the change of rate

• is the actual success value.

3.6.3 EXAMPLE PROBLEMS

Problem1:

Solve the Poisson distribution where =8 and = 12

Step1: Given =8

= 12

e= 2.718

Step2: Formula is

Step3:

=( 2.718) -8

= 0.000335

Step4:

=(8)12

= 68719476736

Step5: Apply the values

=

=

=

= 0.048

Solution: The Poisson distribution is 0.048

Problem2:


Step1: Given =9

= 11


e= 2.718

Step2: Formula is

Step3:

=( 2.718) -9

= 0.0001234

Step4:

=(9)11

= 31381059609


=

=

=

= 0.097

Solution: The Poisson distribution is 0.097

Problem3:


Step1: Given =6

= 12

e= 2.718

Step2: Formula is

Step3:

=( 2.718) -6

=0.00247875

Step4:

=(6)12


= 2176782336


=

=

=

= 0.011

Solution: The Poisson distribution is 0.011.

Practice Problems for Poisson Distribution

Problem1:

Solve the Poisson distribution where =22 = 26

Solution:

The answer of the Poisson distribution where =22 = 26 is 0.055

Problem2:


Solution:


Problem3:


Solution:

The answer of the Poisson distribution where =18 = 21 is 0.068.

Problem4:


Solution:



Study Notes

Assessment

1. The number of pizza orders received at a pizza place follows a Poisson model with a

mean rate of 7 per hour.

a. What is the probability that the pizza shop goes more than 1/2hour between orders?

b. If it has been 1 hour since the last order, what is the probability that an order arrives

in less than 15 minutes?

2. A pizza shop makes deliveries, and the time to make the delivery follows a uniform

distribution between 20 and 35 (minutes): f(x) = 1/15 for 20 < x < 35.

a. Find the average delivery time and the standard deviation of the delivery times.

b. According to Chebyshev's theorem, at least 75% of the delivery times must be

between what two values?

c. On each trip, the supervisor of the drivers gives a bonus of $0.10 for each minute

below 35. For example, if a driver takes 28 minutes, that is a $0.70 bonus. What is

the average bonus per trip?

Discussion

Discuss how Poisson Distribution is different from Probability and Binomial Distribution?


3.7 Normal Distribution

The distribution characterised by the continuous property is termed as normal

distribution. After a vast study, it has been concluded that data collected from the various

fields of science viz., Meteorology, Agriculture, Bioscience, Physics etc, fits the characters of

normal distribution.

Based on the consistency property of the variables of the data series, attempts have

made to evolve the mathematical models highlighting such patterns of distribution to

facilitate the investigation. According to the Statistical Theory, there are three fundamental

distributions viz., Normal, Binomial and the Poisson. The normal distribution is important

amongst them.

The normal distribution, also called the Gaussian distribution, is an important family

of continuous probability distributions, applicable in many fields as mentioned above. Every

constituent member of the family may be defined by two parametres viz., scale and

location.

If we calculate a certain statistic, i.e. parametres are the constants characterising the

population, from X1, X2, X3, ………………………. Xn, then it is found that the mean of above

series is normally distributed and when n tends mass value, the distribution of all the

statistics tends towards the normal.

The importance of the normal distribution as a model of quantitative phenomena lies

in its usage in the natural, social and behavioural sciences. Many measurements, ranging

from psychological to physical phenomena, can be approximated, to varying degrees, by

adopting the normal distribution. While the mechanisms underlying these phenomena are

often unknown, the use of the normal model can be theoretically justified by assuming that

many small, independent effects are additively contributing to each observation. The normal

distribution is also important for its relationship to least-squares estimation. It is believed to

be one of the simplest and oldest methods of statistical estimation.

The normal distribution was first introduced by Abraham de Moivre in 1733, while

the method of least squares was introduced by Legendre in 1805.

The standard normal distribution is the normal distribution with the value of a mean

equal to nil or zero and a variance of one (continuous lined curve in the graph). The

researcher, Carl F. Gauss associated this set of distributions when he analysed astronomical

data and defined the equation of its probability density function. Universally, it is termed as

the bell curve because of its graphical shape, which resembles that of a bell shape.


Fig. 3.2: Normal probability density distribution

3.7.1 DEFINITION OF NORMAL DISTRIBUTION

The simplest case of a normal distribution is known as the standard normal

distribution. It is described by the probability density function

The constant in this expression ensures that the total area under the curve

ϕ(x) is equal to one and 1⁄2 in the exponent makes the “width” of the curve (measured as

half of the distance between the inflection points of the curve) also equal to one. It is

traditional in statistics to denote this function with the Greek letter ϕ (phi), whereas density

functions for all other distributions are usually denoted with letters ƒ or p. The alternative

glyph φ is also used quite often, however, within this article we reserve “φ” to denote

characteristic functions.

More generally, a normal distribution results from exponentiating a quadratic

function (just as an exponential distribution results from exponentiating a linear function):

This yields the classic “bell curve” shape (provided that a < 0 so that the quadratic

function is concave). Notice that f(x) > 0 everywhere. One can adjust a to control the

“width” of the bell, then adjust b to move the central peak of the bell along the x-axis and

finally adjust c to control the “height” of the bell. For f(x) to be a true probability density

function over R, one must choose c such that (which is only possible

when a < 0).

Rather than using a, b and c, it is far more common to describe a normal distribution


by its mean μ = −b/(2a) and variance σ2 = −1/(2a). Changing to these new parametres allows

us to rewrite the probability density function in a convenient standard form,

Notice that for a standard normal distribution, μ = 0 and σ2 = 1. The last part of the

equation above shows that any other normal distribution can be regarded as a version of the

standard normal distribution that has been stretched horizontally by a factor σ and then

translated rightward by a distance μ. Thus, μ specifies the position of the bell curve’s central

peak and σ specifies the “width” of the bell curve.

The parameter μ is at the same time the mean, the median and the mode of the

normal distribution. The parametre σ2 is called the variance; as for any random variable, it

describes how concentrated the distribution is around its mean. The square root of σ2 is

called the standard deviation and is the width of the density function.

The normal distribution is usually denoted by N(μ, σ2). Commonly the letter N is

written in calligraphic font (typed as \mathcal{N} in LaTeX). Thus, when a random variable X

is distributed normally with mean μ and variance σ2, we write

3.7.2 EQUATION OF NORMAL DISTRIBUTION

In the normal distribution pattern, the area under a frequency curve represents the

total number of observations.

Let us assume this area under frequency curve is unity. Then the equation for the

normal curve will be as follows:

Where;

µ and σ are the parameters of the normal distribution

µ = mean of the population under study

σ = standard deviation of the population under study

The above mentioned equation of the normal distribution defines the Y of any value

of X located in between ± ∞ .

1 X - µµµµ

2

- -- ------

1 2 σσσσ Y = ------- e σσσσ √√√√2ππππ


Hence, the form corresponding to the N frequency of the corresponding normal

curve will be

3.7.3 PROPERTIES OF NORMAL DISTRIBUTION

The normal distribution is the standard type of distribution. The following are its

important properties:

• The distribution curve of normal distribution shows symmetrical nature about the mean

(µ) and falls rapidly on either side, tailing off asymptotically to the X – axis in both

directions.

• As mentioned in property one, the X – axis of normal distribution curve is tangent to the

curve of infinity.

• In normal distribution studies, there are only two parametres. They are the mean (µ) and

the standard deviation (σ) of the population

• In normal distribution of population, the relationship among the measures of central

tendency will be as follows:

Here,

Mean = Median = Mode = µ

• Under the normal distribution condition, the first and the third moment about the mean

are zero

µ1 = 0 and µ3 = 0

• The second moment of normal distribution about the mean is equal to the variance (σ 2),

i.e. squared standard deviation.

µ2 = σ 2

• The fourth moment about the mean in normal distribution is 3σ 4

i.e. µ 4 = 3σ 4

• In normal distribution of the population the β1 = 0 and β2 = 3

• Differentiating the equation of the normal distribution

1 X - µµµµ

2

- -- ------

N 2 σσσσ Y = -------- e σσσσ √√√√2ππππ


By differentiating the equation of the normal curve twice with respect to X and

representing the derivatives by ‘Y’

we gets

1

Y’ = − ----- (X -µ) Y

σ2

and

1 X - µ 2

Y” = − ----- 1 − --------- Y

σ2 σ

• The range estimates are as follows:

� The range of µ ± σ includes about 68 % of the observations

� The range of µ ± 2σ includes about 95 % of the observations

� The range of µ ± 3σ includes about 99% of the observations

• The normal distribution shows the property that the sum and differences of normally

distributed variables are also distributed normally.

3.7.4 NORMAL VARIABLE AND NORMAL CURVE

A random variable X, whose distribution has the shape of a normal curve is called a

normal random variable.

1 X - µµµµ

2

- -- ------

1 2 σσσσ Y = ------- e σσσσ √√√√2ππππ


Fig. 3.3: Normal curve

Normal Curve

This random variable X is said to be normally distributed with mean μ and standard

deviation σ if its probability distribution is given by

3.7.5 THE STANDARD NORMAL DISTRIBUTION

To simplify matters, let us standardise our normal curve, with a mean of zero and a

standard deviation of 1 unit.

If we have the standardised situation of μ = 0 and σ = 1, then we have:

Fig. 3.4: Standard normal curve

Standard Normal Curve μ = 0, σ = 1

We can transform all the observations of any normal random variable X with mean μ


and variance σ to a new set of observations of another normal random variable Z with mean

0 and variance 1 by using the following transformation:

We can see this in the following example:

Example

Say μ = 2 and σ = 1/3 in a normal distribution.

The graph of the normal distribution is as follows:

Fig. 3.5: Normal distribution

μ = 2, σ = 1/3

The following graph represents the same information but it has been standardised so

that μ = 0 and σ = 1:

μ = 0, σ = 1


The two graphs have different μ and σ but have the same shape (if we alter the

axes).

The new distribution of the normal random variable Z with mean 0 and variance 1 (or


standard deviation 1) is called a standard normal distribution. Standardising the distribution

like this makes it much easier to calculate probabilities.

If we have mean μ and standard deviation σ, then

Since all the values of X falling between x1 and x2 have corresponding Z values

between z1 and z2, it means:

The area under the X curve between X = x1 and X = x2 equals:

The area under the Z curve between Z = z1 and Z = z2.

Hence, we have the following equivalent probabilities:

P(x1<X<x2) = P(z1<Z<z2)

Example

Considering our example above, where μ = 2, σ = 1/3, then

One-half standard deviation = σ/2 = 1/6 and

Two standard deviations = 2σ = 2/3

So s.d. to 2 s.d. to the right of μ = 2 will be represented by the area from

to . This area is graphed as follows:


μ = 2, σ = 1/3

The area above is exactly the same as the area z1 = 0.5 to z2 = 2in the standard

normal curve:


μ = 0, σ = 1


Percentages of the Area under the Standard Normal Curve

A graph of this standardised (mean 0 and variance 1) normal curve is shown.


In this graph, we have indicated the areas between the regions as follows:

-1 ≤ Z ≤ 168.27%

-2 ≤ Z ≤ 2 95.45%

-3 ≤ Z ≤ 3 99.73%

This means that 68.27% of the scores lie within 1 standard deviation of the mean.

This comes from:


Also, 95.45% of the scores lie within 2 standard deviations of the mean.

This comes from:

Finally, 99.73% of the scores lie within 3 standard deviations of the mean.

This comes from:

The total area from -∞ <z< ∞ is 1.

3.7.6 THE Z-TABLE

The areas under the curve bounded by the ordinates z = 0 and any positive value of z

are found in the z-Table. From this table, the area under the standard normal curve between

any two ordinates can be found by using the symmetry of the curve about z = 0.

EXAMPLE 1

Find the area under the standard normal curve for the following, using the z-table. Sketch

each one.

a. betweenz = 0 and z = 0.78

b. betweenz = -0.56 and z = 0

c. betweenz = -0.43 and z = 0.78

d. betweenz = 0.44 and z = 1.50

e. to the right of z = -1.33.

EXAMPLE 2

Find the following probabilities:

a. P(Z> 1.06)

b. P(Z< -2.15)

c. P(1.06 <Z< 4.00)

d. P(-1.06 <Z< 4.00)

EXAMPLE 3

It was found that the mean length of 100 parts produced by a lathe was 20.05 mm,

with a standard deviation of 0.02 mm. Find the probability that a part selected at random


would have a length:

a. Between 20.03 mm and 20.08 mm

b. Between 20.06 mm and 20.07 mm

c. Less than 20.01 mm

d. Greater than 20.09 mm.

EXAMPLE 4

A company pays its employees an average wage of $3.25 an hour with a standard

deviation of 60 cents. If the wages are normally distributed approximately, determine:

a. The proportion of the workers getting wages between $2.75 and $3.69 an hour

b. The minimum wage of the highest 5%

Study Notes

Assessment

1. What are the Properties of Normal Distribution?

2. Explain Normal Variable and Normal Curve.

3. What is z-Table?

Discussion

The average life of a certain type of motor is 10 years, with a standard deviation of 2 years.

If the manufacturer is willing to replace only 3% of the motors that fail, how long a

guarantee should he offer? Assume that the lives of the motors follow a normal

distribution.


3.8 Exponential Distribution

In Probability Theory and statistics, the exponential distributions (a.k.a.

negative exponential distributions) are a class of continuous probability

distributions. They describe the times between events in a Poisson process, i.e. a

process in which events occur continuously and independently at a constant

average rate.

Fig. 3.10: Exponential Probability density function

Fig. 3.11: Cumulative distribution function


• Probability density function

The probability density function (pdf) of an exponential distribution is

Here λ > 0 is the parameter of the distribution and is often called the rate parameter.

The distribution is supported on the interval [0, ∞). If a random variable X has this

distribution, we write X ~ Exp(λ).

• Cumulative distribution function

The cumulative distribution function is given by

• Alternative parameterisation

A commonly used alternative parameterisation is to define the probability density

function (pdf) of an exponential distribution as

where β > 0 is a scale parameter of the distribution and is the reciprocal of the rate

parameter, λ, defined above. In this specification, β is a survival parameter in the sense that

if a random variable X is the duration of time that a given biological or mechanical system

manages to survive and X ~ Exponential(β) then E[X] = β. That is to say, the expected

duration of survival of the system is β units of time. The parameterisation involving the

"rate" parameter arises in the context of events arriving at a rate λ, when the time between

events (which might be modelled using an exponential distribution) has a mean of β = λ−1

.

The alternative specification is sometimes more convenient than the one given

above and some authors will use it as a standard definition. This alternative specification is

not used here. Unfortunately, this gives rise to a notational ambiguity. In general, the reader

must check which of these two specifications is being used if an author writes

"X ~ Exponential(λ)", since either the notation in the previous (using λ) or the notation in this

section (here, using β to avoid confusion) could be intended.


3.8.1 PROPERTIES OF EXPONENTIAL DISTRIBUTION

1. Mean, variance and median

The mean or expected value of an exponentially distributed random variable X with

rate parameter λ is given by

In light of the examples given above, this makes sense: if you receive phone calls at

an average rate of 2 per hour, then you can expect to wait half an hour for every call.

The variance of X is given by

The median of X is given by

where ln refers to the natural logarithm. Thus, the absolute difference between the

mean and median is

in accordance with the median-mean inequality.

2. Memorylessness

An important property of the exponential distribution is that it is memoryless. This

means that if a random variable T is exponentially distributed, its conditional probability

obeys

This says that the conditional probability that we need to wait, for example, more

than another 10 seconds before the first arrival, given that the first arrival has not yet

happened after 30 seconds, is equal to the initial probability that we need to wait more than

10 seconds for the first arrival. Thus, if we waited for 30 seconds and the first arrival did not

happen (T > 30), the probability that we will need to wait another 10 seconds for the first

arrival (T > 30 + 10) is the same as the initial probability that we need to wait more than 10

seconds for the first arrival (T > 10). This is often misunderstood by students taking courses

on probability: the fact that Pr(T > 40 | T > 30) = Pr(T > 10) does not mean that the events


T > 40 and T > 30 are independent.

To summarise: "memorylessness" of the probability distribution of the waiting time T

until the first arrival means

It does not mean

(That would be independence. These two events are not independent.)

The exponential distributions and the geometric distributions are the only

memoryless probability distributions.

The exponential distribution is consequently also necessarily the only continuous

probability distribution that has a constant failure rate.

3. Quartiles

The quartile function (inverse cumulative distribution function) for Exponential(λ) is

for 0 ≤ p < 1. The quartiles are therefore:

first quartile

median

third quartile

4. Kullback–Leibler divergence

The directed Kullback–Leibler divergence between Exp(λ0) ('true' distribution) and

Exp(λ) ('approximating' distribution) is given by

5. Maximum entropy distribution

Among all continuous probability distributions with support [0,∞) and mean μ, the


exponential distribution with λ = 1/μ has the largest entropy.

6. Distribution of the minimum of exponential random variables

Let X1, ..., Xn be independent exponentially distributed random variables with rate

parameters λ1, ..., λn. Then

is also exponentially distributed, with parameter

This can be seen by considering the complementary cumulative distribution function:

The index of the variable which achieves the minimum is distributed according to the

law

Note that

is not exponentially distributed.

3.8.2 OCCURRENCE AND APPLICATIONS

The exponential distribution occurs naturally when describing the lengths of the

inter-arrival times in a homogeneous Poisson process.

The exponential distribution may be viewed as a continuous counterpart of the

geometric distribution, which describes the number of Bernoulli trials necessary for a

discrete process to change state. In contrast, the exponential distribution describes the time

for a continuous process to change state.

In real-world scenarios, the assumption of a constant rate (or probability per unit

time) is rarely satisfied. For example, the rate of incoming phone calls differs according to

the time of day. On the other hand, if we focus on a time interval during which the rate is

roughly constant, such as from 2 to 4 p.m. during work days, the exponential distribution

can be used as a good approximate model for the time until the next phone call arrives.


Similar caveats apply to the following examples, which yield approximately exponentially

distributed variables:

• The time until a radioactive particle decays or the time between clicks of a Geiger

counter

• The time it takes before your next telephone call

• The time until default (on payment to company debt holders) in reduced form credit risk

modelling

Exponential variables can also be used to model situations where certain events

occur with a constant probability per unit length, such as the distance between mutations

on a DNA strand or between roadkills on a given road.

In Queuing Theory, the service times of agents in a system (e.g. how long it takes for

a bank teller etc. to serve a customer) are often modelled as exponentially distributed

variables. (The inter-arrival of customers, for instance, in a system is typically modelled by

the Poisson distribution in most management science textbooks.) The length of a process

that can be thought of as a sequence of several independent tasks is better modelled by a

variable following the Erlang distribution (which is the distribution of the sum of several

independent exponentially distributed variables).

Reliability Theory and reliability engineering also make extensive use of the

exponential distribution. As a result of the memoryless property of this distribution, it is

well-suited to model the constant hazard rate portion of the bathtub curve used in reliability

theory. It is also very convenient because it is easy to add failure rates in a reliability model.

The exponential distribution is however, not appropriate to model the overall lifetime of

organisms or technical devices because the "failure rates" here are not constant: more

failures occur for very young and for very old systems.

In physics, if you observe a gas at a fixed temperature and pressure in a uniform

gravitational field, the heights of the various molecules also follow an approximate

exponential distribution.


Study Notes

Assessment

1. What do you mean by Exponential Distribution?

2. What are the properties of Exponential Distribution?

Discussion

Discuss occurrence and application of Exponential Distribution.

3.9 Summary

Frequency Distribution: In statistics, a frequency distribution is a tabulation of the

values that one or more variables take in a sample. Each entry in the table contains the

frequency or count of the occurrences of values within a particular group or interval. In this

way, the table summarises the distribution of values in the sample.

Probability Theory: Probability theory is the branch of mathematics concerned with

analysis of random phenomena. Simply put, probability is the ratio of the number of

favourable cases to that of total number of equally likely or possible cases.

Probability Distribution: Probability distribution identifies either the probability of

each value of a random variable (when the variable is discrete) or the probability of the

value falling within a particular interval (when the variable is continuous).


Bayes' Theorem: In probability theory and applications, Bayes' theorem shows the

relation between two conditional probabilities, which are the reverse of each other. This

theorem is named for Thomas Bayes and often called Bayes' law or Bayes' rule. Bayes'

theorem expresses the conditional probability, or "posterior probability", of a hypothesis H

(i.e. its probability after evidence E is observed) in terms of the "prior probability" of H, the

prior probability of E, and the conditional probability of E given H.

Binomial Distribution: The binomial distribution is the discrete probability

distribution of the number of successes in a sequence of n independent yes/no experiments,

each of which yields success with probability p.

Poisson Distribution: The Poisson distribution is a discrete probability distribution

that expresses the probability of a number of events occurring in a fixed period of time if

these events occur with a known average rate and independently of the time since the last

event.

Normal Distribution: The distribution characterised by the continuous property is

termed as normal distribution. The normal distribution, also called the Gaussian distribution,

is an important family of continuous probability distributions, applicable in many fields.

Exponential Distribution: The exponential distributions are a class of continuous

probability distributions. They describe the times between events in a Poisson process, i.e. a

process in which events occur continuously and independently at a constant average rate.


Exercises

1. A random sample of 15 people is taken from a population in which 40% favour a

particular political stand. What is the probability that exactly 6 individuals in the sample

favour this political stand?

a. 0.4000

b. 0.5000

c. 0.2066

d. 0.0041

2. A normal distribution has a mean of 20 and a standard deviation of 4. Find the Z scores

for the following numbers: (a) 28 (b) 18 (c) 10 (d) 23

3. If scores are normally distributed with a mean of 35 and a standard deviation of 10, what

percent of the scores is: (a) greater than 34? (b) smaller than 42? (c) between 28 and 34?


4. According to Financial Executive (July/August 1993) disability causes 48% of all mortgage

foreclosures. Given that 20 mortgage foreclosures are audited by a large lending

institution, what is the probability that less than 8 foreclosures are due to a disability?

5. Ninety percent of the trees planted by a landscaping firm survive. What is the probability

that of the next 13 trees planted:

a. at most ten will survive?

b. at least ten will survive?

c. exactly ten will survive?

Short Notes

a. Properties of Poisson distribution

b. Probability theory

c. Constructing frequency distribution table

d. Normal distribution

e. Applications of probability distributions

f. Bayes' Theorem



2. Business Statistics by Examples, Terry, Sineich, Collier MacMillian Publishers, 1990




1996



Assignment

• A doctor has decided to prescribe two new drugs to 200 heart patients as follows : 50 get

drug A, 50 get drug B and 100 get both the drugs A and B. The 200 patients were chosen

so that each had an 80% chance of having a heart attack if given neither drug. Drug A

reduces the probability of having of a heart attack by 35 %, drug B reduces the

probability by 20% and the two drugs when taken together work independently. If a

randomly selected patient in the programme has a heart attack, what is the probability

that he is given both the drugs? (0.4177)

• Suppose that weights of bags of potato chips coming from a factory follow a normal

distribution with mean 12.8 ounces and standard deviation .6 ounces. If the

manufacturer wants to keep the mean at 12.8 ounces but adjust the standard deviation

so that only 1% of the bags weigh less than 12 ounces, how much does he/she need to

make that standard deviation?

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Unit 4 Correlation, Regression and Time Series

Learning Outcome


• Explain correlation

• Learn the concepts and calculation of correlation

• Identify probable error and standard error in correlation

• Study regression analysis

• Locate standard error in regression

• Elucidate on the pitfalls in regression and regression application

• Define time series

• Characterise Analysis and Models Of Time Series


1. 1st Reading: It will need 3 Hrs for reading a unit


unit




Content Map

4.1 Introduction

4.2 Correlation

4.2.1 Correlation Analysis


4.2.2 Definition of Correlation

4.2.3 Concept of Correlation

4.2.4 Assumptions of Correlation Analysis

4.2.5 Measurement of Correlation

4.2.6 Coefficient of Concurrent Deviation

4.2.7 Probable Error in Correlation (P.E.)

4.2.8 Standard Error

4.2.9 Coefficient of Determination

4.3 Regression Analysis

4.3.1 Properties of Regression Coefficient

4.3.2 Standard Error Estimate

4.3.3 Pitfalls associated with Regressions

4.3.4 Real World Applications using IT tools

4.4 Time Series

4.4.1 Analysis

4.4.2 Models

4.4.3 Notations

4.4.4 Conditions

4.4.5 What is Moving Average or Smoothing Techniques?

4.5 Summary


4.7 Further Reading


4.1 Introduction

Correlation and dependence are any of a broad class of statistical relationships

between two or more random variables or observed data values.

Familiar examples of dependent phenomena include: the correlation between the

physical statures of parents and their offspring and the correlation between the demand for

a product and its price. Correlations are useful because they can indicate a predictive

relationship that can be exploited in practice. For example, an electrical utility may produce

less power on a mild day based on the correlation between electricity demand and weather.

Correlations can also suggest possible causal or mechanistic relationships; however,

statistical dependence is not sufficient to demonstrate the presence of such a relationship.

Formally, dependence refers to any situation in which random variables do not

satisfy a mathematical condition of probabilistic independence. In general statistical usage,

although correlation or co-relation can refer to any departure of two or more random

variables from independence, it most commonly refers to a more specialised type of

relationship between mean values. There are several correlation coefficients, often denoted

ρ or r, measuring the degree of correlation. The most common of these is the Pearson

correlation coefficient, which is mainly sensitive to a linear relationship between two

variables. Other correlation coefficients have been developed to be more robust than the

Pearson correlation or more sensitive to nonlinear relationships.

Regression analysis refers to the techniques used for modelling and analysis of

numerical data consisting of values of a dependent variable (response variable) and of one

or more independent variables (explanatory variables). The dependent variable in the

regression equation is modelled as a function of the independent variables, corresponding

parametres ("constants") and an error term. The error term is treated as a random variable.

It represents unexplained variation in the dependent variable. The parametres are estimated

in order to give a "best fit" of the data. Most commonly, the best fit is evaluated by using the

least squares method, although other criteria have also been used.

Regression can be used for prediction (including forecasting of time-series data),

inference, hypothesis testing and modelling of causal relationships. These uses of regression

rely heavily on the underlying assumptions being satisfied. Regression analysis has been

criticised as being misused for these purposes in many cases, where the appropriate

assumptions cannot be verified to hold. One factor contributing to the misuse of regression

is that it can take considerably more skill to criticise a model than to fit a model.

In statistics, signal processing, econometrics and mathematical finance, a time series


is a sequence of data points, measured typically at successive times spaced at uniform time

intervals. Examples of time series are the daily closing value of the Dow Jones index or the

annual flow of volume of the Nile river at Aswan. Time series analysis comprises methods for

analysing time series data, in order to extract meaningful statistics and other characteristics

of the data. Time series forecasting is the use of a model to forecast future events based on

known past events: to predict data points before they are measured. An example of time

series forecasting in econometrics is predicting the opening price of a stock based on its past

performance.

4.2 Correlation

Correlation is the tendency towards interrelation variation. The measure of such a

tendency is the degree to which the two variables are interrelated and is measured by a

coefficient that is called coefficient of correlation. It gives the degree of association

between the variables.

4.2.1 CORRELATION ANALYSIS

The correlation expresses rates between the groups but not between individual

items. The relationship between two variables is not functional.

Correlation analysis is a statistical procedure by which we can determine the degree

of association or relationship between two or more variables. The amount of correlation in

a data is measured as a coefficient of correlation, which is denoted by r.

4.2.2 DEFINITION OF CORRELATION

The method that is used to find a relationship between two variables (a quantified

bivariate data) is called correlation analysis.

Croxton and Cowden defined correlation as, “The relationship is of quantitative

nature. The appropriate statistical tool for discovering and measuring the relationship and

expressing it in brief formula is known as correlation".

As per A.M. Tuttle, “Correlation is an analysis of the co-variation between two or

more variables".

4.2.3 CONCEPT OF CORRELATION

The relationship between two variables such that a change in one variable results in

a positive or negative change in the other variable and also, a greater change in one variable

results in corresponding greater or smaller change in the other variable is known as

correlation.


The coefficient of correlation between two variables x, y is generally defined by r and

rxy or r(x, y) or r.

There are two types of distribution:

Univariate distribution: In this case, there is only one variable. For example, the

height of students in a class.

Bivariate distribution: In this case, there are two variables such as height and weight

of the students in a class.

Frequency: - Let (xi, yj), i = 1, 2, 3…….m

j = 1,2,3….n, be a bivariate distribution.

If the pair (xi, yj) occurs fi j times than fi j is called the frequency of the pair

(x, y) then the total frequency N= ∑=

m

1i

∑=

n

1j

(fij)

Covariance: The corresponding values of the two variables x and y on the given set of

n unit of observation is given by the pair

(x1 y1), (x2, y2), (x3,y3),……… (xn,yn)

Covariance of x, y, cov(x, y) =

[(x1 - x ) (y1- y ) + (x2 - x ) (y2- y )… (xn - x ) (yn- y )] / n

= 1/n ∑=

n

1i

(xn - x ) (yn- y )

Where x and y are mean of x1 and y1

The above formula for calculation of covariance is difficult and complicated. An

easier method of calculation is:

cov (x, y) = 1/n [∑=

n

1i

xiyj -1/n (∑=

n

1i

xi) (∑=

n

1j

yj)]

Example:

Calculate the covariance of the following pairs of observation of two variables x and y

(1, 6), (2, 9), (3, 6), (4, 7), (5, 8)

Solution:

Σ xi = 1+2+3+4+5 = 15


Σ yj = 6+9+6+7+8 = 36

Σ xi yj = 6+18+18+28+40 = 110

Cov (x1y) = 1/n [∑=

n

1i

xiyj -1/n (∑=

n

1i

xi) (∑=

n

1j

yj)]

= 1

5 [110 -

1

5 (15)(36)]

= 1

5 (110 - 108) =

2

5 = 0.4

Types of Correlation

1. Positive Correlation

If the value of two variables deviates in the same direction as if an increase in one

variable (x) increases the other variable (y), then the correlation is positive or direct. The

height and weight of a growing child can be taken as an example. This is also called linear

correlation.

Fig. 4.1: Positive Correlation

2. Negative Correlation or Inverse Correlation

When two variable x and y deviate in the opposite direction, then the correlation is

said to be negative or inverse.

Fig. 4.2: Negative Correlation


3. No Correlation

If the points that are plotted on the graph are scattered, then there is no correlation

between x and y.

Fig. 4.3: No Correlation

4.2.4 ASSUMPTIONS OF CORRELATION ANALYSIS

Correlation analysis makes the following assumptions:

• The correlation coefficient r is only appropriate for measuring the degree of relationship

between variables that are linearly related to the points to fall along about an imaginary

straight line that passes through the cluster of points.

• The variables are random variables and are measured on either an interval or a ratio

scale.

• The two variables follow bivariate normal distribution for any given values of x, y.

4.2.5 MEASUREMENT OF CORRELATION

1. Karl Pearson’s Method of Correlation (1857 – 1936)

This method is used for measuring linear relationship between two variables (series).

Pearson’s coefficient between two variables (x, y) is denoted by r (x, y) or r(xy) or simply r.

This is also known as product moment correlation coefficient. It is the ratio of the co-

variance [cov (x, y)] to product of standard deviation of x and y. It is given as:

cov(x,y)γ=

σx.σy

σ = standard déviation.

Now for n pairs of observation (x1 y1) (x2 y2)…… (xn, yn)


(I) Cov (x, y) = 1/n ∑ (x - x ) (y - y )

σ1 = 21/nΣ(x-x)

σ2 = 21/nΣ(y-y)

∴ γ = ∑ (x - x ) (y - y )

∑ − )( xx2 ∑

(y - y ) 2

= ∑(dx. dy)/ √∑(dx)2 ∑ (dy)

2 )without σ1 σ2

(II) Also by Direct method

U = 2 2 2 2

nΣxy-(Σx)(Σy)

[nΣx (Σx) )[nΣy (Σ{y) ]

r (x, y) can be written r xy or simply r

Also,

r (x, y) = cor (x, y)

= (r) r (van x) (van (y))

Examples:

1. Calculate the coefficient of correlation for the following data:

(1,2) (2,4) (3,8) (4,7) (5, 10) (6,5) (7, 14) (8, 16) (9, 2) (10, 20)

Solution:

n = 10

Σ x1 = 1+ 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55

Σ y1 = 2 + 4 + 8 + 7 + 10 + 5 + 14 + 16 + 2 + 20 = 88

Σ x5 = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385

Σy5 = 1 + 16 + 64 + 49 + 100 + 25 + 196 + 256 + 4 + 400 = 1114

Σ (x1 y1) = 2 + 8 + 24 + 28 + 50 + 30 + 98 + 128 + 18 + 200 = 586


x = 55

10 = 5.5 y =

88

10 = 8

γ = n Σ x1y1 - (Σ x1) (Σ y1)

√ [[n Σ x12 - (Σ x1)2)] [n Σ y1

2 - (Σ{y1)2] ]

= 10 X 586 - 55 X 88

√10 X 385 - 55 X 55 (10 X 1114 - (88)2)

=1020

825 3396X =

1020

2801700

= 1020

1673.5 = 0.61 (approx)

2. Given below are the monthly incomes and savings of 10 employees of a company.

Calculate the correlation coefficient.

Employee 1 2 3 4 5 6 7 8 9 10

Monthly

Income

780 360 980 250 750 820 900 620 650 390

Net saving 84 51 91 60 66 62 86 58 53 47

Solution:

ξ = 6500

65010

= ; γ = 660

6610

=

No. x y x = x- ξ y = y - γ x5 y5 x y

1 780 84 130 18 16900 324 2340

2 360 51 -290 -15 84100 225 4350

3 980 91 330 25 108900 625 8250

4 250 60 -400 -6 160000 36 2400

5 750 68 100 2 10000 4 200

6 820 62 170 -4 28900 16 -680

7 900 86 250 20 62500 400 500


8 620 58 -30 -8 900 64 240

9 650 53 0 -13 0 169 0

10 390 47 -260 -19 67600 361 4610

Total 6500 660 0 0 539800 2224 27040

r = 1/n (x - x ) (y - y ) / √1/n ∑ (x - x )2 1/n ∑(y - y )2

= ∑xy / √x2 y2 = 27040/ √537800 X 2224 = 0.78 approx.

The value γ indicates a high degree of association between the variables x and y.

From this we can derive that higher the income, higher will be the savings.

2. Spearman’s Rank Correlation

The coefficient of rank correlation is denoted by R.

This is applied to a problem where there is no quantitative data.

Then,

Coefficient of correlation is given by the

R = 1 - 2

2

6

( 1)

D

n n

Σ−

D2 = Square of the difference of corresponding ranks, n = number of paired

observations.

Example:

The ranking of 10 students in Statistics and Accountancy are as follows:

Statistics 3 5 8 4 7 10 2 1 6 9

Accountancy 6 4 9 8 1 2 3 10 5 7

Find the coefficient of rank correlation.

Solution:

Rank x Rank y D D5

3 6 -3 9

5 4 1 1


8 9 -1 1

4 8 -4 16

7 1 6 36

10 2 8 64

2 3 1 1

1 10 9 81

6 5 1 1

9 7 2 4

Total 214

Now R = 2

2

1 6

( 1)

D

n n

− Σ=

− 2

1 6 214

10(10 1)

X−=

−=

2

1 6 214

10(10 1)

X−−

=1 6 214

990

X−

= 1- 1.3 = -0.3

Rank Correlation Coefficient

When the ranks are not given

1. Assign the rank highest first and the lowest last on both x and y

2. Find the rank difference (D), then D2

3. Apply formula as done earlier.

Example:

Calculate the coefficient of correlation from the following data by the method of rank

correlation.

x : 75 88 95 70 60 80 81 50

y : 120 134 150 115 110 140 142 100

x - Assign the highest first

95, 88, 81, 80, 75, 70, 60, 50.


y - 150, 142, 140, 134, 120, 115, 110, 100

X Rank x y Rank y Rank

differences

D2

75 5 120 5 0 0

88 2 134 4 -2 4

95 1 150 1 0 0

70 6 115 6 0 0

60 7 110 7 0 0

80 4 140 3 1 1

81 3 142 2 1 1

50 8 100 8 0

ΣD2 = 6

Coefficient of rank correlation is given by

R = 2

2

1 6

( 1)

D

n n

− Σ

− n = 8

D2 = 4= 1 6 4

8(64 1)

X−−

= 1- 1

21 =

20

21 = 0.9

Merits of rank correlation coefficient:

The merits of rank correlation coefficient can be given as follows:

• Easy to understand and calculate.

• Useful for qualitative data such as beauty honesty etc.

• Useful when qualitative data is not given.

Limitations of rank correlation coefficient:

The limitations of rank correlation coefficient can be given as follows:


• Cannot be used for grouped frequency distribution.

• Not as accurate as Karl Pearson’s coefficient of correlation.

• Cannot be used for more than 30 numbers of items as it consumes time.

• Cannot be used as a continuous series.

4.2.6 COEFFICIENT OF CONCURRENT DEVIATION

The underlying principle in the coefficient of concurrent deviation is as follows:

“If the short term fluctuations of two series are correlated, positively their deviation

would be concurrent and the curves would move in the same direction indicating positive

correlation between the series".

The coefficient of concurrent deviation is given by:

γc = + nc2 − /n

c = number of pairs of concurrent deviations

n = number of pairs of deviations, which one less than actual numbers N = n

= N -1

Example: Calculate the coefficient of correlation by concurrent deviation method.

Price : 1 4 3 5 5 8 10 10 11 15

Demand: 100 80 80 60 58 50 40 40 35 30

Solution:

Price

(X) Cx Demand

(y)

Cy Cx Cy

1 0 100 --

4 -- 80 -- +

3 -- 80 -- +

5 -- 60 -- +


5 -- 58 -- +

8 -- 50 -- +

10 -- 40 -- +

10 -- 40 -- +

11 -- 35 -- +

15 -- 30 -- +

C = 9

Here n = N -1 = 10 -1 = 9

γc = nc2 −−+

/n = 102X9 −−+

/10

ΣPc = + 0.89

Merits:

The merits of coefficient of concurrent deviation can be given as follows:

• Simple to understand and compute

• It is useful for short term fluctuations

Limitations:

The limitations of coefficient of concurrent deviation can be given as follows:

• Not useful for long term range

• Does not differentiate between small and big variations

• The results are a rough indicator of the presence or absence of correlation

4.2.7 PROBABLE ERROR IN CORRELATION (P.E.)

Probable error of the coefficient of correlation (P.E.) can be given as:

= 0.6745 X2 2(1 ) 2 (1 )

3

r r

n n

− −= −

This helps in interpreting its value. This is subject to error of sampling.


Properties of P.E.:

The properties of probable error in correlation can be given as:

• If r = 6 X PE then it is not significant.

• If, r ≤ 6 X PE then it is significant and correlation exists.

• Correlation of the population r ± D P.E. This P.E. is used for testing the reliability value of

r.

Conditions for use of P.E.

The conditions for the use of probable error in correlation can be given as:

• The sample taken should be unbiased and the individual items must be independent.

• The whole data is symmetrical and gives a normal frequency curve (bell shaped).

• The measure of P.E. must be calculated from the sample.

• The items in two series should not be independent of each other.

4.2.8 STANDARD ERROR

Standard error can be calculated using the following formula:

21SE=

r

n

−

r = Coefficient of Correlation

n = number of observations in pairs.

4.2.9 COEFFICIENT OF DETERMINATION

It is the square of the coefficient correlation

= r2, where r = coefficient of correlation

For example, if r = 0.2, r2 = 0.04.

Example:

If n = 25 and P.E. = 0.072 find the values of (i) r and (ii) standard error of

2(1 )r P.E.=0.6745

r

n

−×

n = 25

r =?


PE = 0.072

20.6745 (1 )0.072 =

25

r× −

20.6745 (1 )0.072 =

5

r× −

2 0.07235(1-r ) =

0.6745=

0.360

0.6745=

360

674.5= 0.533

1 – r2 = 0.533

r2 = 1- 0.533 = 0.467

0.467r = = 0.6828

Standard error 21 r

SEn

−= n = 25

1-r2 = 0.533 (given above)

0.533

5= = 0.1066

Study Notes

Assessment

1. Explain concept of Correlation

2. What are the types of Correlation

3. What are the assumptions of correlation analysis

4. What is Probable Error and Standard Error.


Discussion

It is assumed that achievement test scores should be correlated with student's classroom

performance. One would expect that students who consistently perform well in the

classroom (tests, quizes, etc.) would also perform well on a standardized achievement test

(0 - 100 with 100 indicating high achievement). A teacher decides to examine this

hypothesis. At the end of the academic year, she computes a correlation between the

students achievement test scores (she purposefully did not look at this data until after she

submitted students grades) and the overall g.p.a. for each student computed over the

entire year. The data for her class are provided below.

Achievement G.P.A.

98 3.6

96 2.7

94 3.1

88 4.0

91 3.2

77 3.0

86 3.8

71 2.6

59 3.0

63 2.2

84 1.7

79 3.1

75 2.6

72 2.9

86 2.4

85 3.4

71 2.8

93 3.7

90 3.2

62 1.6

1. Compute the correlation coefficient.

2. What does this statistic mean concerning the relationship between achievement

test prformance and g.p.a.?


3. What percent of the variability is accounted for by the relationship between the two

variables and what does this statistic mean?

4. What would be the slope and y-intercept for a regression line based on this data?

5. If a student scored a 93 on the achievement test, what would be their predicted

G.P.A.? If they scored a 74? A 88?

[Answers:

1. r = .524127623 or .52

2. There is a moderate correlation between achievement test performance and g.p.a. As

the achievement test scores go up, the g.p.a.s tend to increase as well and vice versa.

3. r2 = .27 The percent a variability is relatively low. Only 27 percent of the achievement

test performance is related to the g.p.a (and vice versa). Seventy-three percent of the

variability is left unexplained.

4. The slope would be .028430629 and the y-intercept would be .62711903.

5. 3.27; 2.73; 3.13]

4.3 Regression Analysis

We have seen that correlation studies the relationship between two variables X and

Y. If the value of one variable is given and we want to estimate the value of the other

variable, we use the regression method; regression means to go back. In statistics, the term

“regression” is used to denote backward tendency, which means to go back to average or

normal.

Regression analysis is used for estimating or predicting these unknown values of a

variable (called as dependent variable) from the known values of the other variable (called

as independent variable). This is done through the regression line. This describes the

average relationship between the variable x and y.

Regression is simple (using two variables) or multiple and the relationship could be

linear (indicated by a straight line) or non linear.

There are two types of variables in regression analysis : (1) Dependent variable (2)

Independent variable

Dependent variable: The variable, whose value is influenced or is to be predicted is

called the dependent variable.


Independent variable: The variable, which influences the value or is used for

prediction is called the independent variable. The independent variable is also known as

regression or predictor explanatory, while the dependent variable is called a regressed or

explained variable.

Regression equation of y on x

Straight line equation = y = a + b x (y on x) where a and b are constant representing

the Y intercept and the slope of the line respectively. a and b are obtained by solving the

normal equations

Σy = n a + b Σx

and

x y = aΣx + bΣx2

obtained from n given pairs of observations for y and x. We use the form

y - y = byx (x - x )

where byx is the regression. Coefficient of y on x which is given by

byx = Cov (x1 y) = γ σy

or 2 2

( )( )

( ) ( )yx

n xy x yb

n x x

Σ − Σ Σ=

Σ − Σ

Direct Method

Regression equation of x and y

It is expressed as x = a + b y

Σ x = n a + b Σ y

and

Σ x y = a Σ y + b Σ y2

A more convenient form

x - x = bxy (y - y )

Where, bxy is the regression coefficient of x on y which is given by

2 2

( , )xy

Cov x y y x dxdyb

y y dy

σσ σ

Σ= = =

Σ

where dx = x- x


dy = y- y

or

2 2

( )( )

( ( )xy

n xy x yb

n y y

Σ − Σ Σ=

Σ − Σ

4.3.1 PROPERTIES OF REGRESSION COEFFICIENTS

1. Regression coefficients are not symmetric (byx ≠ bxy) unlike the correlation coefficients

[(rxy) = (ryx) = r]

2. Both regression coefficient bxy and byx have the same sign

3. r2 = bxy.byx ∴r = correlation coefficient between x and y

xy yxr b b∴ = ± ⋅ where r has the same sign ( + or -) as that of bxy and by

4.3.2 STANDARD ERROR ESTIMATE

It measures deviation (dispersion) of the central values about the regression line and

is given by

Syx = (standard error of estimate y for given x)

= unexplained error / n = )(e

yy∑ − 2/n

y is the actual

ye is estimated value for given x

Also, Syx = σy 1-r2

The standard error of estimate of x for given y as Sxy

2

eΣ(x-x )

n

Where, x is the actual value and xe is the estimated value of y.

Also, Sxy = σx 21 r−

Standard error of estimate measures the accuracy of the estimated figures. Smaller

its value, better are the estimates and hence more representative is the regression line.

Example:

1. The following data gives the experience of machine operators and their performance

ratings given by the number of goods turned out per 100 pieces.


Operator 1 2 3 4 5 6 7 8

Experience (x) 16 12 18 4 3 10 5 12

Ratings (y) 87 88 89 68 78 80 75 83

Calculate the regression line of performance ratings on experience and estimate the

probable performance if an operator has 7 years of experience.

n = 8

We have x =x

n

Σ=

80

8 = 10

y = y

n

Σ=

81

8=81

Let us create a table.

X Y dx=x-10 dy(y-81) dx2 dy

2 dx. dy x

2 y

2 Xy

16 87 6 6 36 36 36 256 7569 1392

12 88 2 7 4 49 14 144 7744 1056

18 89 8 8 64 64 64 324 792 1602

4 68 -6 -13 36 169 78 16 4624 272

3 78 -7 -3 49 9 21 9 6884 234

10 80 0 -1 0 36 0 100 6400 800

5 75 -5 -6 25 36 30 25 5625 375

12 83 2 2 4 4 4 144 6889 996

80 648 0 0 218 318 247 1018 53676 6727

∴ byx = Σ dxdy = 247 = 1.133

Σdx2 218

By direct method:


byx = 2 2 2

nΣxy-Σx)(Σy) 8X6727-80X648 1976= = =1.133

nΣx -(Σx ) 8X1018-(80) 1744

Equation of regression lines on x is

y - y = byx (x- x )

y - 81 = 1.133 (x - 10)

y - 81 = 1.133x - 11.33

y = 1.133 x + 81 - 11.33

= 1.133x + 69.67 Ans.

If experience is 7 years, the probable performance will be,

x = 7

y = 1.133 x + 69.67 = 1.133x 7 + 69.67 = 7.991 + 69.67 = 77.66 Ans.

2. From the following data

Mean X=36 Y=85

Standard deviation 11 8

Correlation coefficient between x and y is 0.66. Find the regression equation x and y

hence estimate the value of x when y = 75.

Solution:

Given x = 36, y = 85,

σx = 11 σy = 8

r = 0.66

bxy = γ x

y

σσ

= 0.66 X 11

8= 0.908

The regression equation 3on y

x - x = bxy (y- y )

x - 36 = 0.908 (y-85)

x- 36 = 0.908 y - 77.180

x = 0.908 y - 77.180 + 36 = 0.908 y - 41.180


When Y = 75, then x will be

X = 0.90 8 x 75 - 41.180 = 26.92

4.3.3 PITFALLS ASSOCIATED WITH REGRESSION

Also, to the extent that there is a non-linear relationship between the two variables

to be correlated, correlation will not understand the relationship.

Correlation can also be misleading average. The relationship varies depending on the

value of the independent variable. (Lack of homoscedacity)

Homoscedacity: This means that variance around the regression line is the same for

all values of the predictor variable x. The plot shows a violation of this assumption. For the

lower values, the points are all very close to the regression line. For higher values on the x-

axis, there is much more variability around the regression line.

Fig. 4.4: Variance around the regression line

4.3.4 REAL WORLD APPLICATION USING IT TOOLS

Use of statistical methods has undergone a dramatic change as computers and

powerful calculators have emerged in everyday business environments. Companies can

store and manipulate collections of data so that once formidable statistical calculations are

now reduced to a few keystrokes. Sophisticated Windows software allows users to merely

specify the type of analysis required and input the necessary data. Some of the IT

(Information Technology) tools are:

• Microsoft Excel: Excel is a spreadsheet program that can be used to access, process,

analyse, display and share information for running a business. Excel continues to make

the existing functionality easier to use while simultaneously offering a wide array of

tools for making more advanced tasks less complex and more intuitive. Excel is not


designed to be a statistical package; however, it does offer a number of built-in

statistical functions and analysis procedures.

• Minitab: Minitab is a widely used Statistical Analysis Package, originally developed in

1972 at Penn State University to help professors teach basic statistics. Over the years,

Minitab has grown into a powerful and accurate, yet easy-to-use, set of statistical tools.

Minitab is used by a number of Fortune 500 companies and by more than 4000 colleges

and universities worldwide.

• SPSS: SPSS is a large-scale statistical software package designed to integrate and

analyze marketing, customer and operational data. The letter SPSS originally meant

“Statistical Package for the Social Scientists”. Today, SPSS provides solutions that

discover what customers want and predict what they will do.

USES OF STATISTICS IN BUSINESS

Modern businesses need many future predictions in comparison to the small

businesses of the past. Small business managers used to solve most of their problems

through personal contacts. Managers in large corporations, however, must try to

summarise and analyse the various data available to them. They do this by modern

statistical methods.

Here are a list of six areas of business that rely on information and techniques:

• Quality improvement: Statistical quality-control procedures can help give assurance of

high product quality and enhance productivity.

• Product planning: Statistical methods are used to analyse economic factors and business

trends and to prepare detailed sales budgets, inventory-control systems and realistic

sales quotas.

• Forecasting: Statistics are used to predict sales, productivity and employment trends.

• Yearly reports: Annual reports for stockholders are based on statistical treatment of

many cost and revenue factors analysed by the business comptroller.

• Personnel management: Statistical procedures are used in areas of age and sex

discrimination lawsuits, performance appraisals and workforce-size planning.

• Market research: Corporations that develop and market products or services use

sophisticated statistical procedures to describe and analyse consumer purchasing

behaviour.


Study Notes

Assessment

1. What are the properties of Regression Coefficients?

2. What are the drawbacks of Regression?

Discussion

Discuss real world application of regression.

4.4 Time Series

Time series data have a natural temporal ordering. This makes time series analysis

distinct from other common data analysis problems, in which there is no natural ordering of

the observations (e.g. explaining people's wages by reference to their education level, where

the individuals' data could be entered in any order). Time series analysis is also distinct from

spatial data analysis, where the observations generally relate to geographical locations (e.g.

accounting for house prices by the location as well as the intrinsic characteristics of the

houses). A time series model will generally reflect the fact that observations close together

in time will be more closely related than observations further apart. In addition, time series

models will often make use of the natural one-way ordering of time so that values for a

given period will be expressed as deriving in some way from past values, rather than from


future values. (Time series: random data plus trend, with best-fit line and different

smoothing).

Methods for time series analyses may be divided into two classes: frequency-domain

methods and time-domain methods. The former include spectral analysis and recently

wavelet analysis, while the latter include auto-correlation and cross-correlation analysis.

Fig. 4.5: Time series

Definition of time series: Time series is an ordered sequence of values of a variable

at equally spaced time intervals. Time series occur frequently when looking at industrial data

Applications: The usage of time series models is twofold:

• Obtain an understanding of the underlying forces and structure that produced the

observed data

• Fit a model and proceed to forecasting, monitoring or even feedback and feedforward

control

Time series analysis is used for many applications such as:

• Economic forecasting

• Sales forecasting

• Budgetary analysis

• Stock market analysis

• Yield projections

• Process and quality control

• Inventory studies

• Workload projections

• Utility studies


• Census analysis

4.4.1 ANALYSIS

There are several types of data analysis available for time series. These are

appropriate for the following different purposes:

General exploration

• Graphical examination of data series

• Autocorrelation analysis to examine serial dependence

• Spectral analysis to examine cyclic behaviour which need not be related to seasonality.

For example, sunspot activity varies over 11 year cycles. Other common examples

include celestial phenomena, weather patterns, neural activity, commodity prices and

economic activity.

Description

• Separation into components representing trend, seasonality, slow and fast variation,

cyclical irregular: see decomposition of time series

• Simple properties of marginal distributions

Prediction and forecasting

• Fully-formed statistical models for stochastic simulation purposes in order to generate

alternative versions of the time series, representing what might happen over non-

specific time-periods in the future.

• Simple or fully-formed statistical models to describe the likely outcome of the time series

in the immediate future, given knowledge of the most recent outcomes (forecasting).

4.4.2 MODELS

Models for time series data can have many forms and represent different stochastic

processes. When modelling variations in the level of a process, three broad classes of

practical importance are the autoregressive (AR) models, the integrated (I) models and the

moving average (MA) models. These three classes depend linearly on previous data points.

Combinations of these ideas produce Autoregressive Moving Average (ARMA) and

Autoregressive Integrated Moving Average (ARIMA) models. The Autoregressive Fractionally

Integrated Moving Average (ARFIMA) model generalises the former three. Extensions of

these classes to deal with vector-valued data are available under the heading of multivariate

time-series models and sometimes the preceding acronyms are extended by including an


initial "V" for "vector". An additional set of extensions of these models is available for use,

where the observed time-series is driven by some "forcing" time-series (which may not have

a causal effect on the observed series): the distinction from the multivariate case is that the

forcing series may be deterministic or under the experimenter's control. For these models,

the acronyms are extended with a final "X" for "exogenous".

Non-linear dependence of the level of a series on previous data points is of interest,

partly because of the possibility of producing a chaotic time series. However, more

importantly, empirical investigations can indicate the advantage of using predictions derived

from non-linear models over those from linear models.

Among other types of non-linear time series models, there are models to represent

the changes of variance along time (heteroskedasticity). These models are called

Autoregressive Conditional Heteroskedasticity (ARCH). Here, changes in variability are

related to or predicted by, recent past values of the observed series. This is in contrast to

other possible representations of locally-varying variability, where the variability might be

modelled as being driven by a separate time-varying process, as in a doubly stochastic

model.

In recent work on model-free analyses, wavelet transform based methods (for

example, locally stationary wavelets and wavelet decomposed neural networks) have gained

favour. Multiscale (often referred to as multiresolution) techniques decompose a given time

series, attempting to illustrate time dependence at multiple scales.

The general representation of an autoregressive model, known as AR(p), is

where the term εt is the source of randomness and is called white noise. It is

assumed to have the following characteristics:

1.

2.

3.

With these assumptions, the process is specified up to second-order moments and

subject to conditions on the coefficients, may be second-order stationary.

If the noise also has a normal distribution, it is called normal white noise (denoted

here by Normal-WN):


In this case, the AR process may be strictly stationary, again subject to conditions on

the coefficients.

Many types of data are collected over time. Stock prices, sales volumes, interest

rates and quality measurements are typical examples. Owing to the sequential nature of the

data, special statistical techniques that account for the dynamic nature of the data are

required.

The following procedures are followed for analysing time series data:

• Descriptive Methods: Time sequence plots, autocorrelation functions, partial

autocorrelation functions, periodograms and cross-correlation functions are all

important tools for characterizing time series data.

• Smoothing: A variety of smoothers are available to estimate the underlying trend in a

time series.

• Seasonal Decomposition: Decomposes time series data into trend, cycle, seasonal and

irregular components and returns seasonally adjusted data if desired.

• Forecasting: Creation of forecasts beyond the end of the data, using trend models,

moving averages, exponential smoothers or ARIMA models.

• Automatic Forecasting: Selects the best forecasting method for a time series by

optimising a specified information criterion.

DESCRIPTIVE METHODS

Characterising a time series involves estimating not only a mean and standard

deviation but also the correlations between observations separated in time. Tools such as

the autocorrelation function are important for displaying the manner in which the past

continues to affect the future. Other tools, such as the periodogram, are useful when the

data contain oscillations at specific frequencies.


Fig. 4.6: Descriptive Methods

SMOOTHING

When a time series contains a large amount of noise, it can be difficult to visualise

any underlying trend. Various linear and nonlinear smoothers are provided to separate the

signal from the noise.

Fig. 4.7: Smoothing

SEASONAL DECOMPOSITION

When the data contain a strong seasonal effect, it is often important to separate the

seasonality from the other components in the time series. This enables one to estimate the

seasonal patterns and to generate seasonally adjusted data.


Fig. 4.8: Seasonal Decomposition

FORECASTING

A common goal of time series analysis is extrapolating past behaviour into the future.

The forecasting procedures include random walks, moving averages, trend models, simple,

linear, quadratic and seasonal exponential smoothing and ARIMA parametric time series

models. Users may compare various models by withholding samples at the end of the time

series for validation purposes.

Fig. 4.9: Forecasting

4.4.3 NOTATIONS

A number of different notations are in use for time-series analysis:


X = {X1, X2, ...}

is a common notation, which specifies a time series X, which is indexed by the natural

numbers. Another common notation is:

Y = {Yt: t∈T}.

4.4.4 CONDITIONS

There are two sets of conditions under which much of the theory is built:

• Stationary process

• Ergodicity

However, ideas of stationarity must be expanded to consider two important ideas:

strict stationarity and second-order stationarity. Both models and applications can be

developed under each of these conditions, although the models in the latter case might be

considered as only partly specified.

In addition, time series analysis can be applied, where the series are seasonally

stationary or non-stationary. Situations where the amplitudes of frequency components

change with time can be dealt with in time-frequency analysis, which makes use of a time–

frequency representation of a time-series or signal.

4.4.5 WHAT IS MOVING AVERAGE OR SMOOTHING TECHNIQUES?

Smoothing data removes random variation and shows trends and cyclic components.

Inherent in the collection of data taken over time is some form of random variation. There

exist methods for reducing or cancelling the effect due to random variation. An often-used

technique in industry is "smoothing". This technique, when properly applied, reveals more

clearly the underlying trend, seasonal and cyclic components.

There are two distinct groups of smoothing methods

• Averaging Methods

• Exponential Smoothing Methods

Taking averages is the simplest way to smooth data. We will first investigate some

averaging methods, such as the "simple" average of all past data.

A manager of a warehouse wants to know how much a typical supplier delivers in

1000 dollar units. He/she takes a sample of 12 suppliers, at random, obtaining the following

results:


Supplier Amount Supplier Amount

1 9 7 11

2 8 8 7

3 9 9 13

4 12 10 9

5 9 11 11

6 12 12 10

The computed mean or average of the data = 10. The manager decides to use this as

the estimate for expenditure of a typical supplier.

Is this a good or bad estimate?

Mean squared error is a way to judge how good a model is. We shall compute the

"mean squared error":

• The "error" = true amount spent minus the estimated amount.

• The "error squared" is the error above, squared.

• The "SSE" is the sum of the squared errors.

• The "MSE" is the mean of the squared errors.

MSE results for example The results are:

Error and Squared Errors

The estimate = 10

Supplier $ Error Error Squared

1 9 -1 1

2 8 -2 4

3 9 -1 1


4 12 2 4

5 9 -1 1

6 12 2 4

7 11 1 1

8 7 -3 9

9 13 3 9

10 9 -1 1

11 11 1 1

12 10 0 0

The SSE = 36 and the MSE = 36/12 = 3. Table of MSE results for example using

different estimates. So how good was the estimator for the amount spent for each supplier?

Let us compare the estimate (10) with the following estimates: 7, 9 and 12. That is, we

estimate that each supplier will spend $7 or $9 or $12.

Performing the same calculations we arrive at:

Estimator 7 9 10 12

SSE 144 48 36 84

MSE 12 4 3 7

The estimator with the smallest MSE is the best. It can be shown mathematically that

the estimator that minimises the MSE for a set of random data is the mean.

The above table shows squared error for the mean for sample data.

Next, we will examine the mean to see how well it predicts net income over time.

The next table gives the income before taxes of a PC manufacturer between 1985

and 1994.


Year $ (millions) Mean Error Squared Error

1985 46.163 48.776 -2.613 6.828

1986 46.998 48.776 -1.778 3.161

1987 47.816 48.776 -0.960 0.922

1988 48.311 48.776 -0.465 0.216

1989 48.758 48.776 -0.018 0.000

1990 49.164 48.776 0.388 0.151

1991 49.548 48.776 0.772 0.596

1992 48.915 48.776 1.139 1.297

1993 50.315 48.776 1.539 2.369

1994 50.768 48.776 1.992 3.968

The MSE = 1.9508.

The mean is not a good estimator when there are trends. The question arises: can we

use the mean to forecast income if we suspect a trend? A look at the graph below shows

clearly that we should not do this.

Fig. 4.10: Graph of income


Average weighs all past observations equally. In summary, we state that

• The "simple" average or mean of all past observations is only a useful estimate for

forecasting when there are no trends. If there are trends, use different estimates that

take the trend into account.

• The average "weighs" all past observations equally. For example, the average of the

values 3, 4, 5 is 4. We know, of course, that an average is computed by adding all the

values and dividing the sum by the number of values. Another way of computing the

average is by adding each value divided by the number of values or

3/3 + 4/3 + 5/3 = 1 + 1.3333 + 1.6667 = 4.

The multiplier 1/3 is called the weight. In general:

The are the weights and of course they sum to 1.

Study Notes

Assessment

What is Time Series? Explain the procedures followed for analsing time series data.

Discussion

What is Smoothing?

What is Forecasting?

Discuss.


4.5 Summary

Definition of Correlation: Croxton and Cowden definition of correlation: “The

relationship is of quantitative nature. The appropriate statistical tool for discovering and

measuring the relationship and expressing it in brief formula is known as correlation".

Correlation Coefficient: Correlation is the tendency towards interrelation variation

and the coefficient of correlation is a measure of such a tendency is the degree to which the

two variables are interrelated and is measured by a coefficient that is called coefficient of

correlation. It gives the degree of association between the variables.

Karl Pearson’s method of correlation: This method is used for measuring the linear

relationship between two variables (series). Pearson’s Coefficient between two variables (x,

y) is denoted by r (x, y) or r(xy) or simply r. This is also known as product moment correlation

coefficient.

σ = standard déviation.

Spearman’s Rank correlation: The coefficient of rank correlation is denoted by R.

This is applied to a problem where there is no quantitative data.

Concurrent deviation: The principle underlying in the coefficient of concurrent

deviation is as follows: “If the short term fluctuations of two series are correlated positively,

their deviation would be concurrent and the curves would move in the same direction

indicating positive correlation between the series”.

Coefficient of determination: It is square of the Coefficient Correlation

= r2, where r = Coefficient of Correlation

Regression: Regression analysis is used for estimating or predicting these unknown

values of a variable (called as dependent variable) from the known values of other (called as

independent variable). This is done through regression line. This describes the average

relationship between the variable x and y.

Properties of Regression Coefficient:

1. Regression coefficients are not symmetric (byx ≠ bxy) unlike the correlation coefficients

[(rxy) = (ryx) = r]

2. Both regression coefficient bxy and byx have the same sign.

3. r2 = bxy.byx ∴r = correlation coefficient between x and y

xy yxr b b∴ = ± ⋅ where r has the same sign (+ or -) as that of bxy and by


Time Series: A time series is a sequence of data points, measured typically at

successive times spaced at uniform time intervals. Examples of time series are the daily

closing value of the Dow Jones index or the annual flow volume of the Nile River at Aswan.

Time Series Analysis: Time series analysis comprises methods for analysing time

series data, in order to extract meaningful statistics and other characteristics of the data.

Time Series Forecasting: Time series forecasting is the use of a model to forecast

future events based on known past events: to predict data points before they are measured.

An example of time series forecasting in econometrics is predicting the opening price of a

stock based on its past performance.

Application of Time Series Analysis: Time series analysis is used for many

applications such as:

• Economic Forecasting

• Sales Forecasting

• Budgetary Analysis

• Stock Market Analysis

• Yield Projections

• Process and Quality Control

• Inventory Studies

• Workload Projections

• Utility Studies

• Census Analysis

Procedures for analysing time series data: Following are the procedures followed for

analysing time series data:

• Descriptive Methods

• Smoothing

• Seasonal Decomposition

• Forecasting

• Automatic Forecasting


4.6 Self Assessment test

Exercises:

1. Calculate Karl Pearson’s Coefficient of Correlation for the data given below taking 66 and

63 are assumed means of x and y respectively.

Height of Husband x (in Inches) 60 62 64 66 68 70 72

Height of Wife y (in Inches) 61 63 63 63 64 65 62

Ans.: γ = 0.939

2. Calculate Coefficient and Correlation and problem error from the following data.

X 1 2 3 4 5 6 7 8 9 10

Y 20 16 14 10 10 9 8 7 6 5

Ans.: - r = 0.95; P.E. (r) = 0.0208

3. Obtain two lines of regression for the following data:

X 43 44 46 40 44 42 45 42 38 40 42 57

Y 29 31 19 18 19 27 27 29 41 30 26 10

Also find the value of the Correlation Coefficient between x and y. (hint):-

r = _ xy xxb Xb+

Ans.: y = -1.22x + 78.67; x = -0.44y + 54.80; r = -0.7326

4. Given Below is the information about advertising and sales.

Advt exp (in Lakhs) Sales (Rs Lakh)

Mean 10 90

S.D. 3 12

Correlation Coefficient = 0.8

a. Calculate two regressions lines

b. Find the likely sales when advertising expenditure is Rs 15 lakhs


c. What should be the advertising expenditure if the company sales target is of Rs. 120

lakhs

(Ans)

a. Y = 3.2 x + 58X = 0.2y - 8,

b. 106 lakh,

c. 10 lakh.

Short Notes

a. Measurement of correlation

b. Standard error

c. Correlation coefficient

d. Regression

e. Real world application using IT tools

f. Time series

4.7 Further Reading






1996



Assignment

1. Calculate Karl Pearson Coefficient of Correlation from the following data:

2. From the following data obtain the two regression equation:-

X 12 4 20 8 16

Y 18 22 10 16 14

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Year 1985 1986 1987 1988 1989 1990 1991 1992

Index of

production

100 102 104 107 105 112 103 99

Number of

unemployed

15 12 13 11 12 12 19 26


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Unit 5 Linear Programming

Learning Outcome


• Define linear programming

• Explain basic concepts and formulation of linear programming

• Apply methods of solving linear programming

• Analyse graphical method & simplex method

• Describe duality theorem


1. 1st


2. 2nd

Reading with understanding: It will need 4 Hrs for reading and understanding a

unit



5. Revision and Further Reading: It is continuous process

Content Map

5.1 Introduction

5.2 Basic Concept of Linear Programming

5.2.1 Basic Concepts

5.2.2 Limitations of Linear Programming

5.3 Formulation of Linear Programming

5.4 Solution Methods

5.4.1 Graphical method


5.4.2 Simplex method

5.5 The Duality Theorem

5.6 Application of Liner Programming

5.7 Summary


5.9 Further Reading


5.1 Introduction

Linear programming (L.P.) is one of most widely and best understood Operation

Research (OR) techniques. L.P. is confined to the allocation of scarce resources among

various activities in an optimal manner. It determines the way to achieve the best outcome

(such as maximum profit or lowest cost) in a given mathematical model and given some list

of requirements represented as linear equations. Linear programming is a considerable field

of optimisation for several reasons.

This technique originated during world war in order to overcome the use of military

resources. After the war, the industries started using this technique for optimal allocation of

their resources. L.P. is always used for minimising cost and maximising profit in

manufacturing industries and various other industries. Many practical problems in

operations research can be expressed as linear programming problems.

5.2 Basic Concept of Linear Programming 5.2.1 BASIC CONCEPTS

• Linearity assumption: The term linearity means straight line or proportional relationship

with x and y. For example, if one machine and one worker produce, say 100 units per

week, then two machines and two workers will produce 200 units per work (doubled).

i.e. there is a linearity between men and machines.

• Process and its level: Conversion of an input into an output is called a conversion

process. In a process, factor of production is used in fixed ratio, depending upon

technology and as such no substitution is possible within a process. There are many

processes available to a firm for production of a product. One process can be substituted

for another. There is, thus, no interference of one with another. Two or more processes

can be used simultaneously. If a product can be produced in two different ways, then

there are two different processes/ activities/ decision variables.

• Criterion function: This is also known as objective function. This states that

determinants of the quantity are either to be maximised or minimised. For example,

revenue is profit, which needs to be maximized or cost is a function which needs to be

minimised. An objective function should include all possible activities with the revenue

(profit) or cost coefficient per unit of production. The goal is to maximise or minimise

this function. In symbolic form, Zx or Z(x) denotes the value of objective function at the X

level of activities. This is the total sum of activities produced at a specified level.

Activities are denoted as j = 1, 2 … n. The revenue or cost coefficient of the j activity is

represented by C j. Thus, Z(X), implies that for X unit of activity, j = 1 may yield a profit


or loss of C j = 2.

• Inequalities (constraints): This is a restriction imposed on decision variable.

• Feasible solutions: Feasible solutions are all those possible solutions, which can be

worked upon under given constraints.

• Optimum solution: Optimum solution is the best of feasible solution.

Linear programming relationship:

L.P. deals with problems, in which the objective function as well as the constraints

can be expressed as linear mathematical functions of the decision variables.

• Direct proportional line 2X-3Y etc.

• Divisibility variable can take fractional value

e.g. p = ax + by + cz (linear function)

p = ax2 + byx + cz (quadratic function)

p = ax3 + bx2y + cx + dz (cubic function)

5.2.2. LIMITATIONS OF LINEAR PROGRAMMING

Important limitations of linear programming are as under:

• There is no guarantee that linear programming will give integer valued solutions. For

instance, a solution may result in producing a fraction / decimal. In such a situation, the

manager will examine the possibility of producing higher or lower product and will take a

decision, which ensures higher profits subject to given constraints. Thus, rounding can

give reasonably good solutions in many cases but in some situations we will get only a

poor answer even by rounding. Then integer programming techniques alone can handle

such unknown.

• Under linear programming approach, uncertainty is not allowed. The linear

programming model operates only when values for costs, constraints, etc are known but

in real life such factors may be unknown.

• The assumption of linearity is another formidable limitation of linear programming. The

objective functions and the constraint functions in the L.P. model are all linear. We are,

thus, dealing with a system that has constant returns to scale. In many situations, the

input-output rate for an activity varies with the activity level. The constraints in real life

situations concerning business and industrial problems are not linearly related to the

variables. In most economic situations, sooner or later, the law of diminishing marginal


returns begins to operate. In this context, it can, however, be stated that non linear

programming techniques are available for dealing with such situations.

• Linear programming will fail to give a solution if management has multiple conflicting

goals. In L.P. model there is only one goal, which is expressed in the objective function,

e.g. maximising the value of the profit function or minimising the cost function. One

should resort to Goal Programming (G.P.) in situations involving multiple goals.

All these limitations of linear programming indicate only one thing: that linear

programming cannot be made use of in all business problems. Although linear programming

is certainly not a panacea for all management and industrial problems, for those problems

where it can be applied, linear programming is considered a very useful and powerful tool.

Study Notes

Assessment

What is Linear Programming?

Explain the concept of Linear Programming?

Discussion

Discuss the limitations of Linear Programming.


5.3 Formulation of Linear Programming

Formulation means expressing a problem in a convenient mathematical form. Let us

discuss this with an example.

Example:

A carpentry firm manufactures tables and chairs. Data given below are the resources

used and unit profit in manufacturing a table and chair. In this problem there are two

resources namely wood and labour, which are required to produce table and chair. The firm

wants to determine the total profit by maximising the quantity to be produced. The problem

is as given below for formulation in the L.P.P. model.

Unit

requirements

Unit

requirements

Amount

Available

Resources Table Chair

Wood (sq ft) 35 30 350

Labour 10 20 150

Unit profit (Rs) 5 10

Formulation Let X1 - number of table to be produced

Let X2 - number of chair to be produced

Objective function:

Total profit consists of the profit derived from selling a table at. Rs. 5/ per table plus

the profit derived by selling a chair at Rs. 10/.

Hence, profit earned by selling tables = 5X1

Profit earned by selling chair = 10 X2

This needs to be maximized

Z maximize 5x1 + 10x2 -- objective function

Maximize Z, where Z = 5x1 + 10x2

Such that

35x1 + 30x2 ≤ 350

10x1 + 20x2 ≤ 150


x1, x2 ≥ 0

Let us discuss constrains

1. Wood available for table and chair

35 X1 + 30 X2 ≤ 350 (available wood)

(≤ less than or equal to)

Second constraint is labour. This can be expressed in the same way.

10 X1 + 20 X2 ≤ 150 (available labour)

In this, we cannot have negative production, i.e. even when the plant is idle. X1 ≥ 0

and X2 ≥ 0. This is called non-negativity constraints.

Problem can be stated as follow,

• Max Z = 5 X1 + 10X2 objective function.

• 35X1 + 30X2 ≤ 300 constraint

• 10X1 + 20X2 ≤ 150 constraint

and

• X1 ≥ 0 and X2 ≥ 0 non negative condition.

Study Notes

Assessment

Explain how can you express Linear Programming in mathematical form. Give suitable

example.


Discussion

Express in mathematical form:

A farmer needs to buy up to 25 cows for a new herd. He can buy either brown

cows (x) at $50 each or black cows (y) at $80 each and he can spend a total of

no more than $1600. He must have at least 9 of each type. On selling the cows he makes a

profit of $50 on each brown cow and $60 on each black cow.

5.4 Solution Methods

L.P. problems can be solved by two methods:

• Graphical method

• Simplex method

5.4.1 GRAPHICAL METHOD

Graphical method of solving L.P. problems involves two decision variables, x1 and x2.

It includes two major steps:

• Determination of the solution space that defines the feasible solution.

• Determination of optimal solution from feasible region

For example, a furniture manufacturing company plans to make two products hardly,

chairs and tables, from its available resources which consist of 400 cubic feet of mahogany

timber and 450 man hours of labour. It knows that to make a chair it requires 5 cubic feet of

timber and 10 man hours and yields a profit of Rs 80/-. To manufacture a table, it requires

20 cubic feet of timber and 15 man hours and yields a profit of Rs. 90/-. The problem is to

determine how many chairs and tables the company can make keeping within the resources

constraints so that is maximises the profit. Formulate L.P.P. model and provide its graphical

solution.

Graphical method for solving L.P.P.

This method has used two variables (X1 and X2)

Let X1 is denoted for chair.


X2 is denoted for table.

Resources Chair Table Amount

Available

Wood 5 20 400

Labour (man hours) 10 15 450

Formulation

5 X1 + 20 X2 ≤ 400

10X1 + 15 X2 ≤ 450

Z max = 45 X1 + 80 X2 subject to

5 X1 + 20 X2 ≤ 400 i.e. X1 + 4X2 ≤ 80 - - - - (a)

10 X1 + 15 X2 ≤ 450 i.e. 2 X1 + 3 X2 ≤ 90 - - -(b)

X1 X2 ≥ 0

Converting

(a) and (b) to equality, we get

X1 + 4 X2 = 80 . . . . .(1)

2X1 + 3X2 = 90 . . . (2)

Step-1. Let us take the equation (1) i.e

X1 + 4 X2 = 80

Put X1 = 0 i.e. 4 X2 = 80 ∴ X2 = 20

Coordinate (0, 20)

Put X2 = 0, X1 = 80 coordinate (80, 0)

Step -2. Let us take the equation (2)

2 X1 + 3 X2 = 90

Put X1 = 0 3 X2 = 90 X2 = 30 coordinate (0, 30)

Put X2 = 0 2 X1 = 90 X1 = 45 coordinate (45, 0)


Plot this point on the graph. Let us take table on X axis

Chair on Y axis

Fig. 5.1:L.P.P graph

Two straight lines intersect at point B. Its coordinates are calculated as follows:

X1 + 4 X2 = 80 (1)

2X1+ 3 X2 = 90 (2)

Multiply equation (1) by 2

2 X1 + 8X2 = 160

2 X1 + 3 X2 = 90

5 X2 = 70 ∴X2 = 14

∴X1 + 4 × 14 = 80

X1 = 24

i.e. point B in the graph

To find the value of the objective function

Max Z= 45 X1 + 80 X2

90

80

70

60

50

40

30

20

10

00

( 0, 20 )

( C-45,0 )

Table

( 0, 30 )

( B-14,24)

C

H

A

I

R

S


Coordinate of

corner point

Objective function

45 X1 + 80 X2 = Z max Value

(0, 0) 45 X0 + 80 X0 = 0 0

(0, 20) 0 + 1600 1600

(24, 14) 45 × 24 + 80 × 14 2220

(45, 0) 45 × 45 + 80 × 0 2025

This maximum profit is obtained at B i.e. 24 chairs and 14 tables and is equal to

2220.

5.4.2 SIMPLEX METHOD

In the graphical solution, a search for optimal solution is limited to only corner point

of feasible solution region. This problem can be solved manually also, with two variables (x

and y) and less number of constraints. For bigger problems, an efficient procedure is

available for getting optimal solution. This is one of the main objectives of simplex method.

This is a systematic procedure called Algorithm. This method moves from one corner point

to another corner point, till optimal solution is obtained, always improving the objective

function. This method was developed by Prof. George B. Dantzig to solve L.P. problems

involving many variables and constraints.

Let us write down the problem, given earlier:

Zmax = 45 X1 + 80 X2

5 X1 + 20 X2 ≤ 400

10 X1 + 15 X2 ≤ 450

X1, X2 ≥ 0

The inequality constraint must be converted into equality constraint so that the

problem can be solved by adding slack variable to each constraint. Each slack variable

represents unused resources (machine capacity or man hour or materials etc).

Slack variable(s) is non negative. This is added to the “less than” type of constraint to

make it an equality. Always remember that slack variable(s) is added to the equation

(resources availability).

The problem can be stated as follows:


Zmax = 45 X1 + 80 X2 + S1 + S2

subject to 5 X1 + 20 X2 + S1 = 400

10 X1 + 15 X2 + S2 = 450

and X1, X2, S1, S2 ≥ 0

The problem can be tabulated as follows:

Key Row Key Column

Cj Contribution/unit 45 80 0 0 Min. ratio

Basic variable. Solution

Variable X1 X2 S1 S2

0 S1 400 5 20 1 0 400/20=20

0 S2 450 10 15 0 1 450/15=30

Contribution Loss/Unit(Z j) 0 0 0 0 Z = 0

Net Contribution (Cj - Zj) 45 80 0 0

Explanation for the above table:

First row of the table indicates the value of Cj, the coefficient of objective function

and indicates contribution per unit to the objective function of each of the variables. The

second row of the table provides column headings for the table.

The first column heading lies of coefficient of the objective function of the current

basic variable. Second column represents the basic variables in the current solution.

The next column with the heading ‘Solution Values’ is the current solution. In the

example, when the solution is at origin the basic variables are the slack variable S1 and S2

These are listed in second column of the table. Referring back to the first column, the

coefficient for these two variables in the objective functions are S1 = 0 and S2 = 0

respectively. If the current solution at the origin is X1 = 0 and X2 = 0, then the solution values

correspond to S1 = 400 and S2 = 450 as shown in the last column of the table. The column

headed by X1, X2, S1, S2 are the efficient of the constraint set.

The Zj row represents the decrease in the value of the objective function that will


result if one unit of the j the value is brought into solution. Hence, the Z j is a objective

function contribution loss per unit and is found out by adding the product, of C j column and

the coefficient in the constraint set, associated uni-responding basic variable. In the

example, contribution loss per unit (Zj) row values is determined as follows:

Contribution loss per unit (Zj) = Addition of (coefficients of Cj column multiplied by

corresponding coefficients of the constraint set.)

Ci Z j = 0 × 30 + 0 × 10 = 0

Similarly, values of Z j of the column can be calculated and shown in the above table

the last row (C j - Zj) represents net contribution per unit and is determined by subtracting

the appropriate Zj value from the corresponding coefficient C j value in the objective

function for that column. The value (C j - Z j) is the difference between the contribution C j

and the lost Zj that result from one unit of X j being produced. To find out the value of the

last row (Cj - Zj), the necessary calculations are shown as below:

Net contribution per unit is (Cj - Zj).

In the example, (Cj - Zj) for X1 column is: (45 – 0) = 45

Similarly, other columns can be calculated as is shown in the table above.

When (Cj - Zj) is positive, that means that there is an improvement possible in the

existing solution. The objective is to maximise profit, therefore consider the column where

contribution per unit is maximum. In this case, the X2 column contribution per unit is Rs.

80/- (maximum). This helps us to know the variable to be entered into the solution in the

beginning. Thus, X2 is the entering variable. The column corresponding in the entering

variable is known as ‘Key Column’.

Since it is decided to enter one variable as the basic variable into the solution basic,

hence one existing variable is to be departed from solution basic and replace entering

variable to be departed is identified by forming the ratios of solution values to physical rates

of substitution of entering variable. Thus, in the example given above, we have

For, S1= 400/20 = 20

S2 = 450/15 = 30

Hence, departing variable is one which is minimum, i.e. S1 is the departing variable in

our example. This procedure guarantees that there is no negative value in the basic

variable. The row corresponding to the departing variable is called ‘Key Row’. The

intersection of the element of Key Row and Key Column is called the Key Element and is


demoted by this (box) in the simplex table.

• The Simplex Method: Minimisation Case

Until now, we have limited the application of the simplex method only to

maximisation problem. Now that you have developed some familiarity and understanding of

the simplex method, we can apply it to a minimisation problem.

Another way to minimise the problem is to convert the problem by multiplying the

objective function by 1. This yields negative solution values whose sign must be reversed for

application. However this approach is not recommended, since a direct solution is more

convenient to use.

• The Simplex Method (Mixed Constraints)

A situation may arise when the constraints are of mixed type. To simplify the

problem, we can change the last constraint of less than or equal to (≤) type into a equality

(=) type in the problem. Its modified formulation of LP problem can now be written as given

below:

Min Z = 60X1 + 80X2

Subject to X2 ≥ 200

X1 ≤ 400

X1 + X2 = 500

And X1 X2 ≥ 0.

The problem can be converted into the standard form by adding slack, surplus and

artificial variables in the set of constraints and assigning appropriate costs to these variables

in the objective function.

• Some Important Points

DEGENERACY

Sometimes, a linear programming problem may have a degenerate situation.

Degeneracy is revealed when a basic variable acquires a zero value (rather than a negative

or positive value) or in the final solution, either the number of basic variables is not equal to

the number of constraints or the number of zero variables does not equal the number of

decision variables. A tie for an existing variable and an arbitrary selection for it usually

precede the instance of degeneracy. If this is resolved by a proper selection of pivot

element, degeneracy can be avoided.


NON-FEASIBLE SOLUTION

A linear programming problem may be unsolved mathematically due to the

contradictory nature of the constraints. Such an instance is referred to as a non-feasible

solution. A solution is also non-feasible if an artificial variable appears in the basis of the

solution purported to be optimal.

UNBOUNDED SOLUTION

If the coefficient of the entering variable is either negative or zero, implying that this

variable can be increased indefinitely without ever violating feasibility, the maximisation

problem has an unbounded solution.

MULTIPLE SOLUTIONS

The optimal solution may not be unique if one of the non-basic variables has a zero

coefficient in the final Zj - Cj row. This implies that bringing this zero coefficient non-basic

variable in the basis will neither increase nor decrease the value of the objective function.

Thus, the problem has an alternate solution, which is also optimal.

Study Notes

Assessment

Complete the sentence by choosing the correct answer the choices given:

linear programming problems are solved with the computer the meaning of the computer

output and linear programming concepts can be gained by analyzing a simple two-variable

problem with the ———method.

a) Linear programming

b) Graphic method

c) 2-Way method

d) All the above


Discussion

Hale Company manufactures products A and B, each of which requires two processes,

grinding and polishing. The contribution margin is $3 for A and $4 for B. A graph showing

the maximum number of units of each product that can be processed in the two

departments identifies the following corner points: A = 0, B = 20; A = 20, B = 10; A = 30, B =

0. What is the combination of A and B that maximizes the total contribution margin?

[Answer: (a = 0, b = 20); $3(0) + $4(20) = $80 CM

(a = 20, b = 10); $3(20) + $4(10) = $100 CM - Maximum CM

(a = 30, b = 0); $3(30) + $4(0) = $90 CM]

5.5 The Duality Theorem

For every linear programming problem, there is another intimately related linear

programming problem referred to as its dual. The duality theorem states that for every

maximisation (or minimisation) problem in linear programming there is a unique similar

problem of minimisation (or maximisation) involving the same data, which describes the

original problem. The original problem is referred to as the ‘primal’. The ‘dual’ of a ‘dual’

problem is the ‘primal’. Further, the maximum feasible value of the primal objective

function equals the minimum feasible value of the dual objective function. This means that

the solutions of the primal and the dual problems are related, which yield several

advantages.

The transformation of a given primal problem into a dual problem involves the

following considerations:

• If the objective of the primal is maximisation, the objective of dual is minimisation.

• The primal has m-constraints, while its dual has m-unknowns.

• The primal has n-unknowns, while its dual has n-constraints.

• The n-coefficients of the objective function of primal (cj) become the n-constant terms of

its dual.

• The n-constant terms of the primal (bi) become the m-constant terms of the objective

function of its dual.


• The coefficients of the variables of the primal (aij) are transformed in their position in

the dual, i.e. they become aij with respect to the position held in the primal.

• The n-variables (Xn) of the primal are replaced by the m new variables (Ym) of its dual.

This change affects the system of restrictions as well as the objective function.

• The sign of the inequalities in the set of restrictions of the primal (≤) is reversed in the

set of restrictions is its dual (/) and vice-versa. Readers should note that while writing

dual of the primal problem, all the given constraints of the primal should first be

changed in an uniform pattern, say of the z type constraints. This can easily be done as

stated below:

If the constraints is 2X1,+X2 /2, it can also be written as -2X1, -X2 ≤ -2. But if the

constraint is an equation such as

5X1 + 10 X2 ≤150, it can be stated in the form

-5X1 - 10X2 ≤ - 150 without changing the meaning of the given equation.

• The sign of the inequalities restricting the variable (≥ x j) non-negative values in the

primal is equal to the inequality sign of the new variable

(≤ - yj) of its dual.

Thus, by application of these considerations,

Maximise:

Z = 2X1 + 3X2

Subject to:

2X1 + X2 ≤20

X1 + 2X2 ≤20

and

X1, X2 ≥ 0

Transforms to its dual as follows:

Minimise:

Zy = 20Y1 + 20Y2

Subject to:

2Y1 + Y2 / 2


Y1 + 2Y2/3

and

Y1, Y2 ≥ 0

It is instructive to note that the simplex method automatically identifies the dual

basic solution. The optimal value of the objective function remains the same as in the primal

problem. Given an optimal solution of the primal problem, the dual variable acquires the

coefficient of the slack variable in the optimal objective function equation as its optimal

value. In view of all this, it is possible to identify the dual solution from the primal solution.

Exercise

A carpenter makes chairs and tables. Processing of these products is done on

machine A and B. A chair requires 2 hours of machine A and B hours of machine B while a

table requires 5 hours of machine A but does not require machine B. Machine A and B are

available for 16 hours and 20 hours per day respectively. Profits gained by the carpenter per

chair and per table are Rs. 20/- and Rs. 100/- respectively. What should be the daily

production of the two products to realize maximum gain? Formulate LPP.

Hint: Maximise Z = 20 x1 + 100 x2

S.t 2x1 + 5x2 ≤16,

6x1 ≤ 20, x1, x2 ≥ 0

Where x1 = no. of chairs, x2 = No. of tables

Study Notes

Assessment

Explain Duality Theorem in your own words.


Discussion

Discuss how Duality Theorem can be applied in real world.

5.6 Application of Linear Programming

Linear programming (LP) is a significant field of optimization for several reasons.

Many practical problems in operations research can be expressed as linear programming

problems. Certain special cases of linear programming, such as network flow problems and

multicommodity flow problems are considered important enough to have generated much

research on specialized algorithms for their solution. A number of algorithms for other types

of optimization problems work by solving LP problems as sub-problems. Historically, ideas

from linear programming have inspired many of the central concepts of optimization theory,

such as duality, decomposition, and the importance of convexity and its generalizations.

Likewise, linear programming is heavily used in microeconomics and company management,

such as planning, production, transportation, technology and other issues. Although the

modern management issues are ever-changing, most companies would like to maximize

profits or minimize costs with limited resources. Therefore, many issues can be

characterized as linear programming problems.

The areas where LPP is applied are as under:

1. Production and Operations Management

In the process industry, a given raw material can often be made into a wide variety of

products. In the oil industry, for example, crude oil is refined into gasoline, kerosene, home-

heating oil, and various grades of engine oil. There are various profit margins for each

product and it becomes important to determine the best product mix. There are several

limitations such as restrictions on the capacities, raw-material availability, demands and

supply, and any government restrictions on the output of certain products. In such cases,

LPP serves as a useful tool in decision making.

2. Human Resources

Human Resources planning problems can also be solved with linear programming.

For example, in telephone industry, the requirement of installer-repair people is seasonal.

The problem is to determine the number of installer-repair staff and line-repair staff to have

on payroll each month; so the total costs of hiring, layoffs, overtime, and regular-time wages


are minimized. With the use of LPP models such problems can be solved.

3. Marketing

The right mix of media publicity in an advertising campaign is essential decision for

marketing team, where linear programming can prove to be useful tool. For example, the

media available are radio, television, and newspapers. The aim is to decide how many

advertisements are to be placed in each medium. Here, the cost of placing an advertisement

varies for various medium. The aim of every department in organisation is minimisation of

the total cost; here the aim of marketing department thus is minimisation of total cost of the

advertising campaign, keeping in mind the constraints.

4. Distribution

Another application of linear programming is in the area of distribution. For example,

there are a specified number of factories that must ship goods to a given number of

warehouses. One factory could make shipments to any number of warehouses. Here, the

cost of shipping one unit of product from a factory to warehouse is important variable. The

main aim is to minimize the total shipping costs. This decision is subject to various

constraints. Keeping in mind all the constraints and aim of the department, LPP is used to

take suitable decision.

The uses of linear programming are not limited to these five areas but allow you to

easily see why linear programming is so important and how it can practically be applied to

many areas of decision-making.

Study Notes

Assessment

Explain the application of Linear Programming


Discussion

Discuss how Linear Programming is useful for Marketing?

5.7 Summary

Linear Programming: Linear programming has become the most orderly used

mathematical technique in solving a variety of problems related with management- from

scheduling, media selection, financial planning to capital budgeting, transportation and

many others. The special characteristic that linear programming always expects is to

maximise or minimise some quantity. One of the main advantages of linear programming is

that it fits strictly with reality. However, it has limitations too. The most important is the

achievement of goals. It fails to give a solution, where the management has multiple goals.

Basic Concepts: The basic concepts are as follows:

• Linearity assumption: The term linearity means straight line or proportional relationship

with x and y

• Process and its level: Conversion of an input into an output is called a conversion

process. In a process, factor of production are used in fixed ratio, of course, depending

upon technology and as such no substitution possible within a process.

• Criterion function: This is also known as objective function. This states that determinants

of the quantity either to be maximised or minimised.

• Inequalities (Constraints): This a restriction imposed on decision variable.

• Feasible solutions: Feasible solutions are all those possible solutions, which can be

worked upon under given constraints.

• Optimum solution: Optimum solution is the best of a feasible solution.

Linear programming relationship: L.P. deals with problems, in which the objective

function, as well as the constraints, can be expressed as linear mathematical functions of the

decision variables.

Formulation of L.P.: Formulation means expressing a problem in a convenient

mathematical form.

Methods of solving: L.P. problems can be solved by two methods: Graphical method

and Simplex method.


Graphical method: Graphical method of solving L.P. problems involves two decision

variables x1 and x2. It includes two major steps:

• Determination of the solution space that defines the feasible solution.

• Determination of optimal solution from feasible region

Simplex method: This was developed by Prof. George B. Denting to solve L.P.

problems. In the graphical solution, a search for optimal solution was limited to only a

corner point of feasible solution region. This problem can be solved manually also, with two

variables (x and y) and lesser number of constraints. For a bigger problem, an efficient

procedure is available for getting optimal solution. This is one of the main objectives of

simplex method. This is a systematic procedure called algorithm. This method moves from

one corner point to another corner point, till optimal solution is obtained, always improving

the objective function.

The Duality Theorem: For every linear programming problem, there is another

intimately related linear programming problem referred to as its dual. The duality theorem

states that for every maximisation (or minimisation) problem in linear programming, there is

a unique similar problem of minimisation (or maximisation) involving the same data, which

describes the original problem. The original problem is referred to as the ‘primal’. The ‘dual’

of a ‘dual’ problem is the ‘primal’.


Broad Questions

1. What is linear programming? List some problems that can be solved with the help of

linear programming. What characteristics must a problem have if linear programming is

to be used?

2. Describe simplex method of solving a linear programming problem. Why is the simplex

method considered superior to the graphic method?

Short Notes

a. Concepts and assumptions of L.P.

b. Limitations of linear programming

c. Multiple and unbounded solutions

d. Graphical method of solving linear programming problem

e. Duality theorem


Numerical Exercises:

1. Maximise: P = 1.4X1 + X9

Subject to: X1≤ 3

2X1 + X2 ≤ 8

3X1 + 4X2 ≤24

and X1/ 0, X2 /0.

2. Maximise: Z = X1 + X2

Subject to: X1 + X2 ≤3

2X1 + 3X2 ≤18

X1 ≤ 6

and X1, X2 > 0

5.9 Further Reading






1996



Assignment

Comment in your own words, how linear programming is an important part of quantitative

techniques.

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Glossary

Business mathematics: The mathematics used by commercial enterprises to record

and manage business operations. Mathematics typically used

in commerce includes elementary arithmetic, such as fractions,

decimals, and percentages, elementary algebra, statistics and

probability

Business Statistics: The science of good decision making in the face of uncertainty

and is used in many disciplines such as financial analysis,

econometrics, auditing, production and operations including

services improvement, and marketing research

Co-efficient: A numerical value (between +1 and -1) that identifies the

strength of the linear relationship between variables. A value

of +1 indicates an exact positive relationship, -1 indicates an

exact inverse relationship, and 0 indicates no predictable

relationship between the variables

Correlation coefficient: A statistical measure referring to the relationship between two

random variables. It is a positive correlation when each

variable tends to increase or decrease as the other does, and a

negative or inverse correlation if one tends to increase as the

other decreases

Function: A function f of a variable x is a rule that assigns to each number

x in the function's domain a single number f(x). The word

"single" in this definition is very important

Markov Chain: Sequence of stochastic events (based on probabilities instead

of certainties) where the current state of a variable or system

is independent of all past states, except the current (present)

state. Movements of stock/share prices, and growth or decline

in a firm's market share, are examples of Markov chains. It is

named after the inventor of Markov analysis, the Russian

mathematician Andrei Andreevich Markov (1856-1922)

Matrices: Flat (two-dimensional) table, in which the elements or entries

appear at the intersections of rows and columns, governed by

certain rules. They are called rectangular array in mathematics


Mean: The average of a numerical set. It is found by dividing the sum

of a set of numbers by the number of members in the set

Median: The value of a numerical set that equally divides the number of

values that is larger and smaller

Mode: The value of a numerical set that appears with the greatest

frequency

Moving Average: moving average is a form of average which has been adjusted

to allow for seasonal or cyclical components of a time series.

Moving average smoothing is a smoothing technique used to

make the long term trends of a time series clearer

Normal Distribution: Also called "bell curve," the normal distribution is the curved

shape of a graph that is highest in the middle and lowest on

the sides

Poisson distribution: The distribution of number of events in a given time, arising

from a Poisson process. This differs from the binomial

distribution in that there is no upper limit, corresponding to

the parameter 'n' of a Binomial Process, to the number of

events which may occur

Probability: The measure of how likely it is for an event to occur. The

probability of an event is always a number between zero and

100%. The meaning (interpretation) of probability is the

subject of theories of probability. However, any rule for

assigning probabilities to events has to satisfy the axioms of

probability

Regression Analysis: A technique used for the modelling and analysis of numerical

data consisting of values of a dependent variable (response

variable) and of one or more independent variables

(explanatory variables)

Sequence: An ordered set, whose elements are usually determined based

on some function of the counting numbers

Series: The sum of the terms of a sequence. Finite sequences and

series have defined first and last terms, whereas infinite

sequences and series continue indefinitely

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