By: Aalia Akram

Surabhi Gupta

Shiwani Awasthi

Jaya Kumari

Purvi Bajpai

PROBABILITY DISTRIBUTION

Brief of Probability:

• The concept originated in seventeenth centaury.

• The formulae and technique were developed by Jacoub Bernoulle, De Moiver, Thomas Bayed and Joseph Lagrange.

• Nineteenth centaury -Pierre Simon, Laplace unified all these ideas and compiled general theory of probability.

• In probability distribution, the variable are distributed according to some definite probability functions. These distributions occupy a place of great prominence in business decision making.

PROBABILITY DISTRIBUTION

Probability distribution is list for all possible occurrences. The probability distribution or the distribution of discrete random variable is a list of distinct values x1 of X together with their associative probabilities,

i.e. f(x1)= P(X=x1)

The numbers represented by f(x) are all between 0 and

Thus we can say that,

‘A listing of the probabilities for every possible value of a random variable is called a Probability Distribution.’

CONCEPT OF PROBABILITY DISTRIBUTION

• The meaning can be made more clearer by following points:

An observed frequency distribution: it is a listing of the observed frequencies of all the outcome of an experiment that actually occurred while performing the experiment.

A probability distribution: it is the listing of the probabilities of all possible outcome that could result if the experiment is performed.

Theoretical frequency distribution is a probability distribution that outcomes are expected to vary. In other words, it enlists the expected values that is observed values multiplied by corresponding probabilities of all out come.

• For example:

There are four coins. These are tossed simultaneously 80 times. The possible results which can be obtained, can be arranged in a specific set called Sample Space (S)

• S= HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, TTH, THTH, THHT, HTTT, THTT, TTHT, TTTH, TTTT

No. Of Heads

(r)

Probability p(r)

p (x)

Expected Frequency

Np=80 p(r)

0

1

2

3

4

1/16

4/16

6/16

4/16

1/16

80*1/16=5

80*4/16=20

80*6/16=30

80*4/16=20

80*1/16=5

Total 1 80

• The Probability Distribution is as follows:

r: 0 1 2 3 4

P(r): 1/16 4/16 6/16 4/16 1/16

• The Theoretical Frequency Distribution is:

r: 0 1 2 3 4

p(r): 5 20 30 20 5

TYPES OF PROBABILITY DISTRIBUTION

• The 4 types of Probability Distribution are:

1. PROBABILITY DISTRIBUTION OF A DISCRETE RANDOM VARIABLE

2. DISCRETE PROBABILITY DISTRIBUTION

o Binomial Distribution

o Poisson Distribution

3. PROBABILITY DISTRIBUTION OF A CONTINUOUS RANDOM VARIABLE

4. CONTINOUS PROBABILITY DISTRBUTION

o Normal Distribution

PROBABILITY DISTRIBUTION OF A DISCRETE RANDOM VARIABLE

• In discrete random variable, there is a probability value assigned to each event. Also we can say it assumes a finite number of values or a random variable which takes countable values. The distribution follows the three rules required of all probability distribution:

1. The events are mutually exclusive and collectively exhaustive

2. The individual probability values are between 0 & 1 inclusive

3. The total of probability values sum to 1.

DISCRETE PROBABILITY DISTRIBUTION

Discrete probability distribution is the probability distribution of a

discrete random variable ‘X’ as function f(x) satisfying the

following conditions:

I. f(x) ≥ 0

II. ∑f(x) = 1

III. (X = x) = f(x)

PROBABILITY DISTRIBUTION OF A CONTINUOUS RANDOM VARIABLE

• Continuous random variables are those which takes place in continuity.

• Examples like the time takes to finish a project, the high temperature of a day etc.

• A random variable is continuous if its range forms an uncountable set of real numbers. It is the one in which the random variable r can assume any value between certain limits a and b. the no. of all possible values is then uncountable infinite.

• With a continuous probability distribution there is a continuous mathematical function that describes the probability distribution. This function is called the probability density function or simply the probability function. It is usually represented by f (r).

CONTINUOUS PROBABILITY DISTRIBUTION-

• Unlike the discrete probability distributions, the concept of associating a probability with each possible value of a continuous random variable is no longer meaningful. In case of continuous random variables, we can only talk about the probability that the random variable falls within a given interval. A continuous probability distribution can be visualized as a discrete probability distribution with a large number of values very close to each other.

PART-2

BINOMIAL DISTRIBUTION

BINOMIAL DISTRIBUTION IS DEVELOPED A SWISS SCIENTIST

James Bernoulli in 1700

DEFINITION

Binomial is a discrete probability distribution expressing the probability of one set of dichotomous alternatives, i.e., success and failure.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; when n = 1, the binomial distribution is a Bernoulli distribution.

Bernoulli Trials It is a process wherein an experiment is performed repeatedly, yielding either a “success” or a ‘failure” in each trial and absolutely no pattern of successes and failures.

PROPERTIES

• It is a discrete distribution,based on binomial theorem , which gives theoretical probability.

• There will be two and only two possible outcomes i. e. either success or failure.

• Each trial is individual and independent. It means that the outcome of any observation is independent of the outcome of any other trial, it neither influences nor be influenced by the outcome of any other trial.

• The binomial coefficients are given by Pascal triangle.

PROPERTIES

• The process is performed under the same conditions for fixed and finite number of trials say ’n’.

• The probability of a success denoted by ‘p’ remains the same/constant from trial to trial. The probability of a failure denoted by ‘q’ is equal to (1-p). If the probability of success is not the same in each trial, we will not have a binomial distribution.

• It depends on the parameter p or q and n.• The shape and location of a binomial distribution

changes as p changes for a given n and n changes for a given p.

HOW BINOMIAL DISTRIBUTION ARISES If a coin is tossed once there are two outcomes, namely, head or tail. The probability of obtaining a head or p = ½ and the probability of obtaining a tail or q= (1-p) = ½. Thus (q+p) =1. These are terms of the binomial (q+p). Similarly, if two coins are tossed simultaneously there are four possible outcomes: A B A B A B A B T T T H H T H H The probabilities corresponding to these outcomes are: T T T H H T H H q q q p p q p p q2 2qp p2

These are the terms of the binomial (q+p) ^2 because (q+p)^ 2 = q^2 + 2qp + p^2

In general in n tosses of a coin the probabilities of the various events (i.e. obtaining 0, 1, 2, 3 ….n heads) are given by the successive terms of the binomial expansion (q+p)^n , which is (q+p)^ n = q^n + nC1q^(n-1)p + nC2q^(n-2)p^2 +………+nCrq^(n-r)p^r + ……..+ p^nThese terms may be listed in the form of a probability distribution table as follows:

From the above binomial expansion, the following general relationship should be noted:

• The number of terms in a binomial expansion is always n+1• The exponents of p and q, for any single term, when added

together, always sum to n.• The exponents of q are n, (n-1), (n-2)…….1, 0. Respectively and

the exponents of p are 0, 2…….. (n-1), n, respectively (note: p0= 1; q0= 1).

• The coefficients for the n+1 terms of the distribution are always symmetrical ascending to the middle of the series and then descending, When n is odd number, n+1 is even and the coefficients of the two central terms are identical.

Probability of obtaining exactly ‘r’ successes in a given numbers of ‘n’ Bernoulli trials is,

f (r) = P (X=r) = ncr p^

r q^(n-r) for r = 0, 1, 2, 3, ……….n

Where,

p = probability of success on a single trial q = probability of failure on a single trial

n = number of (Bernoulli) trials r = number of successes in ‘n’ trials

GENERAL MODEL OF BINOMIAL DISTRIBUTION-

Find out the binomial distribution to be expected by tossing 4 coins 320 times.Solution: - Let X = no. of successes i.e. the no. of heads when 4 coins are tossed n= 4, no. of coins p= Probability of getting a head on a coin = ½ q= 1-p = ½ Possible values of X are: 0 1 2 3 4

The corresponding probabilities are obtained in the expansion (q+p)n:=>(q+p)4 = q4+4C1q3p+4C2q2p2+4C3qp3+p4=>(1/2 +1/2) 4 = (1/2)4+4(1/2)3(1/2)+6(1/2)2(1/2)2+4(1/2)(1/2)3+(1/2)4 = 1/16+4/16+6/16+4/16+1/16Thus, we have X = r : 0 1 2 3 4P (X = r) : 1/16 4/16 6/16 4/16 1/16

Expected frequency = NP (X = r), where n = 320

The required theoretical binomial distribution is as follows:

X = r : 0 1 2 3 4

NP (X=r) : 20 80 120 80 20

When a binomial distribution is to be fitted to observe data, the following procedure is adopted:

• Determine the values of p and q. If one of these values is known, the other can be found out by the simple relationship p = (1-q) and q = (1- p). When p and q are equal, the distribution is symmetrical. Then p and q may be interchanged without altering the value of any term, consequently, terms equidistant from the two ends of the series are equal. If p and q are unequal, the distribution is skewed. If p is less than 0.5, the distribution is positively skewed and when p is more than 0.5, the distribution is negatively skewed

Fitting of Binomial Distribution

• Expand the binomial (q+p). The power n is equal to one less than the number of terms in the expanded binomial. Thus, when n = 2there will be three terms in the binomial. Similarly, when n = 4 there will be five terms.

• Multiply each term of the expanded binomial by N (the total frequency), in order to obtain the expected frequency in each category.

f(x)= P [X=x] = nCx p^x q^(n-x)f(x+1)= P [X=x+1] = nCx+1p^(x+1)q^(n-x-1)f(x+1)/ f(x) = P(n-x)/q(x+1)f(x+1) = P(n-x)/q(x+1)When x = 0,

f(1)= P(n-0)/ q(0+1) f(0)=pnf(0)/q When x = 1, f(2)=p(n-1)/q2 f(1)=(p/q)2n(n-1)f(0)/2!When x=2, F(3)=p(n-1)/q3 f(2)=(p/q)3n(n-1)(n-2)f(0)/3!

and so on.

This formula provides us a very convenient method for fitting the binomial distribution.

The only probability we need to calculate is f(0) which is equal to q^n, where q can be

estimated from the given data.

APPLICATION OF BINOMIAL DISTRIBUTION

• The binomial distribution is applicable only when sampling is from an infinite statistical population with replacement because in these cases the success probability remains constant from trial to trial.

• The binomial distributionis very useful in decision making situation in business.

• One area in which it is very widely applied is quality control.

• Binomial distributionis is considered an important tool in forecasting of the events based on random sampling.

POISSON DISTRIBUTION

PART-3

POISSON DISTRIBUTION DEVELOPED BY FRENCH MATHEMATICIAN

SIMEON DENIS

POISSON

(1781- 1840)

POISSON DISTRIBUTION

A second important discrete probability distribution is the Poisson distribution, named after the French mathematician, Simeon Denis Poisson who published its derivation in 1837.

The distribution is used to describe the behavior of rare events such as the number of accidents on road .number of printing mistakes in a book etc, and has been called “the law of improbable events.” 

THE CHARACTERISTICS OF THE POISSON DISTRIBUTION

• The occurrence of the events is independent.

• Theoretically, an infinite number of occurrences of the event must be possible in the interval.

• The probability of single occurrence of the event in a given interval is proportional to the length of the interval.

• In any infinitesimal (extremely small) portion of interval, the probability of two or more occurrences of the event is negligible.

Conditions of Poisson Distribution

• The variable is discrete

• The numbers of trials i.e. n should be very large

• The probability of success i.e. p is very small

• The probability of success in each trial is constant

• np is constant and finite

Form of Poisson Distribution • It has a single parameter which is the mean of the distribution and is denoted by m.

• The Probability of exactly 0, 1, 2, 3, …….n successes is found by the successive term of the following expansion. *

e-m where

• e = 2.71828. It is the base of the natural system of logarithms, m = mean

• When x = 0,p (0) = Probability of exactly zero success = and so on

• Thus the probabilities of exactly 0,1,2 & 3 success would be

GENERAL FORMULA OF POISSON DISTRIBUTION

• p(r)= Where,

r= 0,1,2,3,4…………..

e= 2.7183(the base of natural logarithms)

m= mean of Poisson distribution= np

• From d above diagram as “m” increases, the distribution shifts to the right. Thus, all Poisson probability distribution are skewed to the right.

Mean and Variance of the Poisson Distribution

• The Mean. The mean of the Poisson distribution is given by

= m

• Thus, the mean of the Poisson distribution is m.

• The Variance. The variance of the Poisson distribution is given by

• Thus, the variance of the Poisson distribution is also equal to m.

FITTING OF A POISSON DISTRIBUTION

• Finding out the theoretical values in a Poisson distribution is very easy. We have to first find out the values of mean & calculate the frequency of 0 successes. Once this is done the other frequencies can be obtained very easily as shown below:-

• P (r) when r = 0 = e -m

• If this probability is multiplied by N or the total number of observation we get the frequency for 0 success.

• Thus frequency of 1 success =N (p0) =

• The frequency of 1 success = N ( =N (

• The frequency of 2 success = N (p2) = N (p1) ×m/2 and so on…..

(Ques) The following table gives the number of days in a 50- day period

during which automobile accidents occurred in a city. Fit Poisson

distribution in the data.

• No. of accidents: 0 1 2 3 4

• No. of days : 21 18 7 3 1

• Solution: Fitting of Poisson distribution

x f fx

0 21 0

1 18 18

2 7 14

3 3 9

4 1 4

N=50 fx=45

• m = xˉ=Σƒ x/N=45/50=0.9

• ƒ (0) = = = 0.4066

• ƒ (1) =m ƒ (0) = (0.9) (0.4066) = 0.3659

• ƒ (2) =m/2ƒ (1) = 0.9/2 (0.3659) = 0.1647

• ƒ (3) =m/3ƒ (2) = 0.9/3 (0.1647) = 0.0494

• ƒ (4) =m/4ƒ (3) = 0.9/4 (0.0494) =0.0111

• In order to fit Poisson distribution, we shall multiply each probability by N, i.e., 50

• Hence, the expected frequencies are:-

• X: 0 1 2 3 4

• F: 0.4066×50 0.3659×50 0.1647×50 0.0494×50 0.0111×50

=20.33 =18.30 =8.24 =2.47 =0.56

Applications of Poisson distribution

Number of typographical errors per page in typed material. Number of deaths in a town by a rare disease. In insurance Problems: Number of casualties. Number of accidents in factory. Number of calls arrived at a switch board. Demand for a product. In biology: Count the number of bacteria. In Physics: Count the number of particles emitted from a

radioactive substance.

NORMAL DISTRIBUTION

PART-4

HISTORY OF NORMAL DISTRIBUTION

It was first developed by Abraham De Moivre (1667-1754) as the limiting form of the binomial model in1733, it was rediscovered by Gauss in 1809 and by Laplace in 1812.

MEANING :-

Normal Distribution is a continuous distribution which is used for determining a suitable mathematical distribution for dealing with quantities whose magnitude is continuously variable.Normal Distribution happens to be most useful theoretical distribution for continuous variables.

The normal distribution is an approximation to binomial Distribution. whether or not p is equal to q, the binomial distribution tends to the form of a continuous curve & when 'n' becomes large at least for the material part of the range.

The limiting frequency curve obtained as ‘n’ becomes large is called the normal frequency curve or simply the normal curve.

EQUATION FOR THE NORMAL DISTRIBUTION

Where: - x = value of continuous random variable µ = mean of the normal random variable e = mathematical constant=2.7183 Л = mathematical constant=3.1416 σ = standard deviation of normal random variable

EQUATION TO A NORMAL CURVE CORRESPONDING TO A PARTICULAR DISTRIBUTION IS THUS GIVEN BY

here, N/σ √2π = max. ordinate (y0) of the normal curve

corresponding to distribution of stated total frequency ‘N’ and stated standard deviation ‘σ’.

GRAPH OF A NORMAL DISTRIBUTION

REMARKS:

1) The normal distribution can have different shapes depending on different values of µ and σ but there is one and only one normal distribution for any given pair of values for µ and σ.

2) Normal distribution is a limiting case of binomial distribution when n --> ∞ and neither ‘p’ nor ‘q’ is very small.

3) Normal distribution is a limiting case of poisson distribution when its mean ‘m’ is large.

4) The two nails of normal distribution extend indefinitely and never touch the horizontal axis.

USE & IMPORTANCE OF NORMAL DISTRIBUTION• Normal distribution is applied for large application in statistical quality control in industry for setting control limits. Even if a variable is not normally distributed, it can sometime be brought to normal form by simple transformation of variable. For example, if distribution of X is skewed, the distribution of √ x might come out to be normal. • As ‘ n’ becomes large, the normal distribution serves as a good approximation for many discrete distribution.• The sampling distribution and test of hypothesis are based upon the assumption that samples have been drawn from a normal population with mean µ and variance σ2.• It has numerous mathematical properties which make it popular and comparatively easy to manipulate.

• Central limit theorem:- according to this theorem, as the

sample size ‘n’ increases, the distribution of x of a random sample taken from practically any population approaches a normal distribution. Thus if samples of large size ‘n’ are drawn from a population i.e. not normally distributed, never the less the successive samples means will form themselves a distribution. i.e. approximately normal.Thus, it is possible to determine the minimum and maximum limits within which the population values lie. • Universal utility.• Basis of sampling & interpretation.• Mathematical relationship between constants.

Characteristics of a normal distribution• Bell shaped • Continuous• Equality of central value • Uni-model• Equal distance of quartiles from median• Relationship between Q.D. & S.D• Asymptotic to the base line• Height

ASSUMPTIONS OF NORMAL DISTRIBUTION

Independent factors- The forces affecting events must be independent of one another.

Numerous factors- The causal forces must be numerous & of equal weight.

Symmetry- The operation of the causal forces must be such that positive deviation from the mean balanced as to magnitude & number by negative deviation from the mean.

Homogeneity- These forces must be the same over the universe from which the observations are drawn.

FITTING OF A NORMAL CURVE

• Two main objectives of fitting a normal curve is to sample data:-

1) To provide a visual device for judging whether or not normal curve is a good fit to a sample data.

2) To use the smoothed normal curve, instead of the irregular curve representing the sample data, to estimate the characteristics of the population.

Methods of fitting:

Method of ordinates

Method of areas

Fitting of normal distribution In order to fit a normal distribution to the given data, we 1st calculate the mean µ and standard deviation σ from the given data. Then the normal curve fitted to the given data is given by; -1/2(x-µ/ σ)2

Y= 1/ σ√2πe

-∞<x<∞

To calculate the expected normal frequency, we 1st find the standard normal variate corresponding to the lower units of each class interval.

Ques:- fit a normal curve to the following data by method of ordinates-

variable frequency

60 - 62 5

63 - 65 18

66 - 68 42

69 - 71 27

72 - 74 8

for fitting a normal curve we need the values of and σ.

variable m f d fd f(d2

60-62 61 5 -2 -10 20

63-65 64 18 -1 -18 18

66-68 67 42 0 0 0

69-71 70 27 +1 +27 27

72-74 73 8 +2 +16 32

N=100 Σ fd=15 Σ fd2 = 97

= A + Σfd / N * i

= 67 + 15 / 100 * 3

= 67.45

σ = √Σfd2/N – (Σfd / N)2 * i

= √97/100 – (15/100)2 * 3

= √0.97 – 0.0225 * 3

= 0.0973 * 3

=2.92

Mean ordinate = 0.399 * Ni/ σ

=0.399 * 100 * 3/ 2.92

=41

Therefore, the height of the maximum ordinate of the point 67.45 on abscissa will be 41.

STANDARD NORMAL DISTRIBUTION It is useful to transform a normally distributed variable in such a form that single tables of areas under the normal curve would be applicable regardless of the unity of the original distribution.Let x be a random variable distributed normally with mean x & standard deviation. Then we define a new random variable z using transformation technique.Formula z = x-x/ σ & x/ σ [x=x-x] Where z=new random variable X=value of x X=mean of x σ =S.D of xThen z is called a standard normal variate & its distribution is called standard normal distribution with mean 0 & standard deviation unity.

PART-5

RELATIONS

Relation between Binomial and Normal Distribution

Binomial distribution can be closely approximated by a normal distribution under the following conditions:

When the number of trials n, is very large, n→∞Neither ‘p’ nor ‘q’ is too close to zero (very small or close to 0.50)

The standard random variable is given by:

Z will follow normal distribution with mean ‘zero’ and variance one.

• Relation between Normal and Poisson Distribution

• Since, there is a relation between normal and binomial distribution, it is expected that there is a relation between normal and Poisson distribution.

• Poisson distribution approaches to a normal distribution with standardized variable i.e.

  ‘m’ increases infinitely (∞), Z will follow normal distribution with mean ‘zero’ and variance ‘one’. 

APPLICATION OF PROBABILITY DISTRIBUTIONS

• Student appearing in the examination.

• Number of telephone calls received at a telephone exchange during a period of time.

• The no. of births taking place in a nursing home.

• t depends on the parameter p,q & n

• The binomial coefficients are given by Pascal.

All the trials are assumed to be independent and each trial has two outcomes namely success and failure. 

BINOMIAL DISTRIBUTION

Poisson Distribution

• Number of traffic arrival such as a ship at a dock, bus at depots, aeroplane at airport.

• In quality control statics such as no. of defective items.

• In insurance problems such as number of accidents in factory.

• Number of calls arrived at a switchboard.

• The no. of defective electric bulbs manufacture by a reputed company.

• The no. of cars passing a certain point in one minute.

• The no. of persons born blind per year in a large city.

• The no. of deaths in a town by a rare disease.

• Demand of a product.

• In biology-counting the no. of bacteria

Normal Distribution

• To check errors in astronomical observations

• To check errors in physical measurements

• To check deviation from benchmark measures quality control.

• Measuring errors in survey studies: census data, market data, economic data etc.

• Check risk in finance: stock market data, portfolio theory, stock returns, option prices etc.

• Universal utility.

• basis of sampling & interpretation

Conclusion

• So far in this presentation topic of probability distribution we have defined a random variable i.e. numerical distribution of the outcome of an experiment and different types of probability distributions like Binomial distribution, Poisson distribution and Normal distribution. By probability distribution we can save our time and effort. Probability distribution helps us in finding the expected value in context to sales and production.

• A discrete random variable will follow a ‘binomial distribution’ if it is the number of successes in ‘n’ independent Bernoulli trials with probability of success constant. ‘Poisson probability distribution’ is a probability distribution showing the probability of X occurrences of an event over a specified interval of time.

• ‘Normal distribution’ is a continuous probability distribution, bell shaped, determined by mean ‘µ’ and standard deviation ‘σ’. Normal distribution is used as a model in real life situation both as a continuous distribution and as an approximation to discrete distributions. Normal distribution is used widely in statistical inferences.