Call Letter for Reliable Techno Designs

8/14/2019 Call Letter for Reliable Techno Designs

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Hi,

----------------------------------------------------------------------------------------------------------------------

To,

Reliable Techno-Designs Pvt. Ltd.

BG/SEI 11/03, MIDC Bhosari,

Near BSNL Office, Pune - 411026Landline:-020-30684519, Mob:- 9922912177

Sub: Interview of the short listed candidate for your requirement ofAsst.Manager/ Deputy Manager Press Tool Designg

Dear Tushar Sir,

We are pleased to send the candidate bearing this letter,

Mr Mangesh C. Girgaonkar

who is short-listed by us for the interview. His CV is attached herewith for yourready reference.

We await your valued feedback at your earliest convenience.

With Best Regards,

Mrs.Swet

ha.F.Hottinavar

HR (Sr.Recruiter)

For Mandar Sants PlacementServices

Random variable

MANDAR SANTSPLACEMENT SERVICES.

VIJIGEESHA, 17, PRASHANT NAGAR,Behind KAKA HALWAI, L.B.S. Road,

Pune - 411030CONTACT:- 020-24332433, 9922435001


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Broadly, there are two types of random variable discrete and continuous. Discreterandom variables take on one of a set of specific values, each with some probabilitygreater than zero. Continuous random variables can be realized with any of a range ofvalues (e.g., a real number between zero and one), and so there are several ranges (e.g. 0to one half) that have a probability greater than zero of occurring.

A random variable has either an associated probability distribution (discrete randomvariable) or probability density function (continuous random variable).

Random Variable

The outcome of an experiment need not be a number, for example, the outcome

when a coin is tossed can be 'heads' or 'tails'. However, we often want to

represent outcomes as numbers. A random variable is a function that associates

a unique numerical value with every outcome of an experiment. The value of the

random variable will vary from trial to trial as the experiment is repeated.

There are two types of random variable - discrete and continuous. A random

variable has either an associated probability distribution (discrete random

variable) or probability density function (continuous random variable).

Examples--A coin is tossed ten times. The random variable X is the number of tails thatare noted. X can only take the values 0, 1, ..., 10, so X is a discrete random variable.

1. A light bulb is burned until it burns out. The random variable Y is its

lifetime in hours. Y can take any positive real value, so Y is a continuous

random variable.

Expected Value

The expected value (or population mean) of a random variable indicates its

average or central value. It is a useful summary value (a number) of the

variable's distribution.

Stating the expected value gives a general impression of the behaviour of somerandom variable without giving full details of its probability distribution (if it is

discrete) or its probability density function (if it is continuous).

Two random variables with the same expected value can have very different

distributions. There are other useful descriptive measures which affect the shape

of the distribution, for example variance.
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The expected value of a random variable X is symbolised by E(X) or .

If X is a discrete random variable with possible values x1, x2, x3, ..., xn, and p(xi)

denotes P(X = xi), then the expected value of X is defined by:

where the elements are summed over all values of the random variable X.

If X is a continuous random variable with probability density function f(x), then the

expected value of X is defined by:

Example

Discrete case : When a die is thrown, each of the possible faces 1, 2, 3, 4, 5, 6

(the xi's) has a probability of 1/6 (the p(xi)'s) of showing. The expected value of

the face showing is therefore:

= E(X) = (1 x 1/6) + (2 x 1/6) + (3 x 1/6) + (4 x 1/6) + (5 x 1/6) + (6 x 1/6)

= 3.5

Notice that, in this case, E(X) is 3.5, which is not a possible value of X.

See also sample mean.

Variance

The (population) variance of a random variable is a non-negative number which

gives an idea of how widely spread the values of the random variable are likely to

be; the larger the variance, the more scattered the observations on average.

Stating the variance gives an impression of how closely concentrated round the

expected value the distribution is; it is a measure of the 'spread' of a distribution

about its average value.

Variance is symbolised by V(X) or Var(X) or

The variance of the random variable X is defined to be:

where E(X) is the expected value of the random variable X.
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Notes

a. the larger the variance, the further that individual values of the random

variable (observations) tend to be from the mean, on average;

b. the smaller the variance, the closer that individual values of the randomvariable (observations) tend to be to the mean, on average;

c. taking the square root of the variance gives the standard deviation, i.e.:

d. the variance and standard deviation of a random variable are always non-

negative.

See also sample variance.

Probability Distribution

The probability distribution of a discrete random variable is a list of probabilities

associated with each of its possible values. It is also sometimes called the

probability function or the probability mass function.

More formally, the probability distribution of a discrete random variable X is a

function which gives the probability p(xi) that the random variable equals xi, for

each value xi:

p(xi) = P(X=xi)

It satisfies the following conditions:

a.

b.

Cumulative Distribution Function
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All random variables (discrete and continuous) have a cumulative distribution

function. It is a function giving the probability that the random variable X is less

than or equal to x, for every value x.

Formally, the cumulative distribution function F(x) is defined to be:

for

For a discrete random variable, the cumulative distribution function is found by

summing up the probabilities as in the example below.

For a continuous random variable, the cumulative distribution function is the

integral of its probability density function.

Example

Discrete case : Suppose a random variable X has the following probability

distribution p(xi):

xi 0 1 2 3 4 5

p(xi) 1/32 5/32 10/32 10/32 5/32 1/32

This is actually a binomial distribution: Bi(5, 0.5) or B(5, 0.5). The cumulative

distribution function F(x) is then:

xi 0 1 2 3 4 5

F(xi) 1/32 6/32 16/32 26/32 31/32 32/32

F(x) does not change at intermediate values. For example:

F(1.3) = F(1) = 6/32

F(2.86) = F(2) = 16/32

Probability Density Function

The probability density function of a continuous random variable is a function

which can be integrated to obtain the probability that the random variable takes a

value in a given interval.


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More formally, the probability density function, f(x), of a continuous random

variable X is the derivative of the cumulative distribution function F(x):

Since it follows that:

If f(x) is a probability density function then it must obey two conditions:

a. that the total probability for all possible values of the continuous random

variable X is 1:

b. that the probability density function can never be negative: f(x) > 0 for all x.

Discrete Random Variable

A discrete random variable is one which may take on only a countable number of

distinct values such as 0, 1, 2, 3, 4, ... Discrete random variables are usually (but

not necessarily) counts. If a random variable can take only a finite number of

distinct values, then it must be discrete. Examples of discrete random variables

include the number of children in a family, the Friday night attendance at a

cinema, the number of patients in a doctor's surgery, the number of defective

light bulbs in a box of ten.

Compare continuous random variable.

Continuous Random Variable

A continuous random variable is one which takes an infinite number of possiblevalues. Continuous random variables are usually measurements. Examples

include height, weight, the amount of sugar in an orange, the time required to run

a mile.
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Compare discrete random variable.

Independent Random Variables

Two random variables X and Y say, are said to be independent if and only if thevalue of X has no influence on the value of Y and vice versa.

The cumulative distribution functions of two independent random variables X and

Y are related by

F(x,y) = G(x).H(y)

where

G(x) and H(y) are the marginal distribution functions of X and Y for all

pairs (x,y).

Knowledge of the value of X does not effect the probability distribution of Y and

vice versa. Thus there is no relationship between the values of independent

random variables.

For continuous independent random variables, their probability density functions

are related by

f(x,y) = g(x).h(y)

where

g(x) and h(y) are the marginal density functions of the random variables X

and Y respectively, for all pairs (x,y).

For discrete independent random variables, their probabilities are related by

P(X = xi ; Y = yj) = P(X = xi).P(Y=yj)

for each pair (xi,yj).

Probability-Probability (P-P) Plot

A probability-probability (P-P) plot is used to see if a given set of data follows

some specified distribution. It should be approximately linear if the specifieddistribution is the correct model.

The probability-probability (P-P) plot is constructed using the theoretical

cumulative distribution function, F(x), of the specified model. The values in the
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sample of data, in order from smallest to largest, are denoted x (1), x(2), ..., x(n). For

i = 1, 2, ....., n, F(x(i)) is plotted against (i-0.5)/n.

Compare quantile-quantile (Q-Q) plot.

Quantile-Quantile (QQ) Plot

A quantile-quantile (Q-Q) plot is used to see if a given set of data follows some

specified distribution. It should be approximately linear if the specified distribution

is the correct model.

The quantile-quantile (Q-Q) plot is constructed using the theoretical cumulative

distribution function, F(x), of the specified model. The values in the sample of

data, in order from smallest to largest, are denoted x(1), x(2), ..., x(n). For i = 1, 2,

....., n, x(i) is plotted against F-1((i-0.5)/n).

Compare probability-probability (P-P) plot.

Normal Distribution

Normal distributions model (some) continuous random variables. Strictly, a

Normal random variable should be capable of assuming any value on the real

line, though this requirement is often waived in practice. For example, height at a

given age for a given gender in a given racial group is adequately described by a

Normal random variable even though heights must be positive.

A continuous random variable X, taking all real values in the range is said

to follow a Normal distribution with parameters and if it has probability

density function

We write

This probability density function (p.d.f.) is a symmetrical, bell-shaped curve,

centred at its expected value . The variance is .
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Many distributions arising in practice can be approximated by a Normal

distribution. Other random variables may be transformed to normality.

The simplest case of the normal distribution, known as the Standard Normal

Distribution, has expected value zero and variance one. This is written as N(0,1).

Examples

Poisson Distribution

Poisson distributions model (some) discrete random variables. Typically, a Poisson

random variable is a count of the number of events that occur in a certain time interval orspatial area. For example, the number of cars passing a fixed point in a 5 minute interval,or the number of calls received by a switchboard during a given period of time.

A discrete random variable X is said to follow a Poisson distribution with parameter m,written X ~ Po(m), if it has probability distribution

where
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x = 0, 1, 2, ..., n

m > 0.

The following requirements must be met:

a. the length of the observation period is fixed in advance;

b. the events occur at a constant average rate;

c. the number of events occurring in disjoint intervals are statistically

independent.

The Poisson distribution has expected value E(X) = m and variance V(X) = m; i.e.

E(X) = V(X) = m.

The Poisson distribution can sometimes be used to approximate the Binomial

distribution with parameters n and p. When the number of observations n is

large, and the success probability p is small, the Bi(n,p) distribution approaches

the Poisson distribution with the parameter given by m = np. This is useful since

the computations involved in calculating binomial probabilities are greatly

reduced.

Examples

Binomial Distribution

Binomial distributions model (some) discrete random variables.
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Typically, a binomial random variable is the number of successes in a series of

trials, for example, the number of 'heads' occurring when a coin is tossed 50

times.

A discrete random variable X is said to follow a Binomial distribution withparameters n and p, written X ~ Bi(n,p) or X ~ B(n,p), if it has probability

distribution

where

x = 0, 1, 2, ......., n

n = 1, 2, 3, .......

p = success probability; 0 < p < 1

The trials must meet the following requirements:

a. the total number of trials is fixed in advance;

b. there are just two outcomes of each trial; success and failure;

c. the outcomes of all the trials are statistically independent;

d. all the trials have the same probability of success.

The Binomial distribution has expected value E(X) = np andvariance V(X) =

np(1-p).

Examples
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Geometric Distribution

Geometric distributions model (some) discrete random variables. Typically, a

Geometric random variable is the number of trials required to obtain the first

failure, for example, the number of tosses of a coin untill the first 'tail' is obtained,

or a process where components from a production line are tested, in turn, until

the first defective item is found.

A discrete random variable X is said to follow a Geometric distribution with

parameter p, written X ~ Ge(p), if it has probability distribution

P(X=x) = px-1

(1-p)x

where

x = 1, 2, 3, ...

p = success probability; 0 < p < 1

The trials must meet the following requirements:

a. the total number of trials is potentially infinite;

b. there are just two outcomes of each trial; success and failure;

c. the outcomes of all the trials are statistically independent;

d. all the trials have the same probability of success.

The Geometric distribution has expected value E(X)= 1/(1-p) and variance

V(X)=p/{(1-p)2}.

The Geometric distribution is related to the Binomial distribution in that both are

based on independent trials in which the probability of success is constant and

equal to p. However, a Geometric random variable is the number of trials until the

first failure, whereas a Binomial random variable is the number of successes in n

trials.
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Examples

Uniform Distribution

Uniform distributions model (some) continuous random variables and (some)

discrete random variables. The values of a uniform random variable are uniformly

distributed over an interval. For example, if buses arrive at a given bus stop every

15 minutes, and you arrive at the bus stop at a random time, the time you wait for

the next bus to arrive could be described by a uniform distribution over the

interval from 0 to 15.

A discrete random variable X is said to follow a Uniform distribution with

parameters a and b, written X ~ Un(a,b), if it has probability distribution

P(X=x) = 1/(b-a)

where

x = 1, 2, 3, ......., n.

A discrete uniform distribution has equal probability at each of its n values.

A continuous random variable X is said to follow a Uniform distribution with

parameters a and b, written X ~ Un(a,b), if its probability density function is

constant within a finite interval [a,b], and zero outside this interval (with a less

than or equal to b).

The Uniform distribution has expected value E(X)=(a+b)/2 and variance {(b-

a)2}/12.
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Example

Central Limit Theorem

The Central Limit Theorem states that whenever a random sample of size n is

taken from anydistribution with mean and variance , then the sample mean

will be approximatelynormally distributed with mean and variance /n. The

larger the value of the sample size n, the better the approximation to the normal.

This is very useful when it comes to inference. For example, it allows us (if the

sample size is fairly large) to use hypothesis tests which assume normality even

if our data appear non-normal. This is because the tests use the sample mean

, which the Central Limit Theorem tells us will be approximately normally

distributed.

Measures of central tendancy: sample mean sample median modeDefinition: if our sample isx1, x2, . . . , xn then

x= sample mean =x1 +x2 + .+xn/n

Mean, Mode, Median, and Standard

Deviation


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measure is the median. The median is the middle score. If we have an even number ofevents we take the average of the two middles. The median is better for describing thetypical value. It is often used for income and home prices.

Example

Suppose you randomly selected 10 house prices in the South Lake Tahoe area. Your areinterested in the typical house price. In $100,000 the prices were

2.7, 2.9, 3.1, 3.4, 3.7, 4.1, 4.3, 4.7, 4.7, 40.8

If we computed the mean, we would say that the average house price is 710,000.Although this number is true, it does not reflect the price for available housing in SouthLake Tahoe. A closer look at the data shows that the house valued at 40.8 x $100,000 =$4.08 million skews the data. Instead, we use the median. Since there is an even numberof outcomes, we take the average of the middle two

3.7 + 4.1= 3.9

2

The median house price is $390,000. This better reflects what house shoppers shouldexpect to spend.

There is an alternative value that also is resistant to outliers. This is called thetrimmed

mean which is the mean after getting rid of the outliers or5% on the top and 5% on thebottom. We can also use the trimmed mean if we are concerned with outliers skewing thedata, however the median is used more often since more people understand it.

Example:

At a ski rental shop data was collected on the number of rentals on each of tenconsecutive Saturdays:

44, 50, 38, 96, 42, 47, 40, 39, 46, 50.

To find the sample mean, add them and divide by 10:

44 + 50 + 38 + 96 + 42 + 47 + 40 + 39 + 46 + 50= 49.2

10


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Notice that the mean value is not a value of the sample.

To find the median, first sort the data:

38, 39, 40, 42, 44, 46, 47, 50, 50, 96

Notice that there are two middle numbers 44 and 46. To find the median we take theaverage of the two.

44 + 46Median = = 45

2

Notice also that the mean is larger than all but three of the data points. The mean isinfluenced by outliers while the median is robust.

Variance, Standard Deviation and Coefficient of Variation

The mean, mode, median, and trimmed mean do a nice job in telling where the center ofthe data set is, but often we are interested in more. For example, a pharmaceuticalengineer develops a new drug that regulates iron in the blood. Suppose she finds out thatthe average sugar content after taking the medication is the optimal level. This does notmean that the drug is effective. There is a possibility that half of the patients havedangerously low sugar content while the other half have dangerously high content.Instead of the drug being an effective regulator, it is a deadly poison. What thepharmacist needs is a measure of how far the data is spread apart. This is what thevariance and standard deviation do. First we show the formulas for these measurements.Then we will go through the steps on how to use the formulas.

We define the variance to be

and thestandard deviation to be

Variance and Standard Deviation: Step by Step


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1. Calculate the mean, x.2. Write a table that subtracts the mean from each observed value.3. Square each of the differences.4. Add this column.5. Divide by n -1 where n is the number of items in the sample This is the

variance.6. To get thestandard deviationwe take the square root of the variance.

Example

The owner of the Ches Tahoe restaurant is interested in how much people spend at therestaurant. He examines 10 randomly selected receipts for parties of four and writesdown the following data.

44, 50, 38, 96, 42, 47, 40, 39, 46, 50

He calculated the mean by adding and dividing by 10 to get

x = 49.2

Below is the table for getting the standard deviation:

x x - 49.2 (x - 49.2 )2

44 -5.2 27.04

50 0.8 0.64

38 11.2 125.44

96 46.8 2190.24

42 -7.2 51.84

47 -2.2 4.84

40 -9.2 84.64

39 -10.2 104.04

46 -3.2 10.24

50 0.8 0.64

Total 2600.4


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Now

2600.4= 288.7

10 - 1

Hence the variance is 289 and the standard deviation is the square root of 289 = 17.

What this means is that most of the patrons probably spend between $32.20 and $66.20.

The sample standard deviation will be denoted by s and thepopulation standard deviation

will be denoted by the Greek letter.

The sample variance will be denoted by s2 and the population variance will be denoted by

2.

The variance and standard deviation describe how spread out the data is. If the data alllies close to the mean, then the standard deviation will be small, while if the data isspread out over a large range of values, s will be large. Having outliers will increase thestandard deviation.

One of the flaws involved with the standard deviation, is that it depends on the units thatare used. One way of handling this difficulty, is called the coefficient of variation whichis the standard deviation divided by the mean times 100%

CV = 100%

In the above example, it is

17100% = 34.6%

49.2

This tells us that the standard deviation of the restaurant bills is 34.6% of the mean.


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arithmetic mean

Definition

Simple average,equalto the sum of all values divided by the number of values.

The geometric mean, inmathematics, is a type ofmeanoraverage, which indicates thecentral tendency or typical value of a set of numbers. It is similar to the arithmetic mean,

which is what most people think of with the word"average," except that instead of adding the set of numbersand then dividing the sum by the count of numbers in the

set, n, the numbers are multiplied and then the nth root of the resultingproduct is taken.

For instance, the geometric mean of two numbers, say 2 and 8, is just the square root (i.e.,the second root) of their product, 16, which is 4. As another example, the geometric meanof1, , and is the cube root (i.e., the third root) of their product (0.125), which is .

teps involved in constructing a frequency

polygon

The most common method of graphingresearch data is to construct a frequencypolygon. The first step in constructing afrequency polygon is to list all scores and totabulate how many subjects received eachscore. Steps involved in constructing afrequency polygon are: 1) list all scores andtabulate how many subjects received each score,

2) place all the scores on a horizontal axis, atequal intervals from lowest score to the highest,3) place the frequencies of scores at equalintervals on the vertical axis, starting with zero,4) for each score, find the point where the scoreintersects with its frequency of occurrence andmake a dot, 5) connect all the dots with straightlines.

In fact, along with the development ofcomputer technology, there are several graphicdesigning software packages that can generalize

the frequency polygon easily, such asSpreadsheet in ClarisWorks and Excel inMicrosoft office.Three measures of central tendency

Measures of central tendency give theresearcher a convenient way of describing a setof data with a single number. Three mostfrequently encountered indices of central

Geometric mean
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tendency are the mode, the median, and themean. The mode is appropriate for themeasurement of nominal data, the median for

the ordinal data, and the mean for the interval

or ratio data.

Quartile deviation: In "research talk", thequartile deviation is one-half of the differencebetween the upper quartile and the lowerquartile in a distribution. In English, the upperquartile is the 75th percentile; it means there are75% scores below than that point. Bysubtracting the lower quartile from the upperquartile and then dividing the result by two, weget a measure of variability. If the quartiledeviation is small the scores are close together,whereas if the quartile deviation is large the

scores are more spread out. The quartiledeviation is a more stable measure of variabilitythan the range and is appropriate whenever themedian is appropriate. Standard Deviation: The standarddeviation is the square root of the variance,which is based on the distance of each scorefrom the mean. It is appropriate when the datarepresent an interval or ratio scale. It is themost stable measure of variability and takes intoaccount each and every score. Measuringstandard deviation is to find out how far eachscore is from the mean, that is, subtracting themean from each score. Steps for calculating thestandard deviation are: 1) Find out N, thenumber of subjects, 2) Calculate the sum of thescores, 3) square each score, 3) Add all thesquares, to get the sum of squares of the scores,4) Square the sum of the scores and divide bythe number of scores (we have a measure ofvariability called variance), 5) Subtract thevariance from the sum of the squares of scoresto get the sum of the squares (SS), 6) divide theSS by N-1. A small standard deviationindicates the scores are close together and alarge standard deviation indicates that the scoresare more spread out.


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0 - 9 1

10 - 19 2

20 - 29 3

30 - 39 4

40 - 49 5

50 - 59 4

60 - 69 3

70 - 79 2

80 - 89 2

90 - 99 1

To construct the histogram, groups are plotted on thex axis and their frequencies on theyaxis. The following is a histogram of the data in the above frequency table.

Histogram

Information Conveyed by Histograms

Histograms are useful data summaries that convey the following information:

The general shape of the frequency distribution (normal, chi-square, etc.)

Symmetry of the distribution and whether it is skewed

Modality - unimodal, bimodal, or multimodal

The histogram of the frequency distribution can be converted to a probability distributionby dividing the tally in each group by the total number of data points to give the relativefrequency.

The shape of the distribution conveys important information such as the probabilitydistribution of the data. In cases in which the distribution is known, a histogram that does


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not fit the distribution may provide clues about a process and measurement problem. Forexample, a histogram that shows a higher than normal frequency in bins near one end andthen a sharp drop-off may indicate that the observer is "helping" the results by classifyingextreme data in the less extreme group.

Yourcontinued donations keep Wikipedia running!

From Wikipedia, the free encyclopedia

Jump to: navigation, search

Probability theory is the branch ofmathematics concerned with analysis ofrandomphenomena.[1] The central objects of probability theory arerandom variables,stochasticprocesses, andevents: mathematical abstractions ofnon-deterministic events or measuredquantities that may either be single occurrences or evolve over time in an apparentlyrandom fashion. Although an individual coin toss or the roll of adieis a random event, ifrepeated many times the sequence of random events will exhibit certain statistical

patterns, which can be studied and predicted. Two representative mathematical resultsdescribing such patterns are the law of large numbers and the central limit theorem.

As a mathematical foundation forstatistics, probability theory is essential to many humanactivities that involve quantitative analysis of large sets of data. Methods of probabilitytheory also apply to description of complex systems given only partial knowledge of theirstate, as in statistical mechanics. A great discovery of twentieth centuryphysics was theprobabilistic nature of physical phenomena at atomic scales, described inquantummechanics.

Probability theory
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Contents

[hide]

1 History

2 Treatment

o

2.1 Discrete probability distributionso 2.2 Continuous probability distributions

o 2.3 Measure-theoretic probability theory

3 Probability distributions

4 Convergence of random variables

5 Law of large numbers

6 Central limit theorem

7 See also

8 References

9 Bibliography

[edit] History

The mathematical theory ofprobability has its roots in attempts to analyse games ofchanceby Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat andBlaise Pascal in the seventeenth century (for example the "problem of points").

Initially, probability theory mainly considered discrete events, and its methods weremainlycombinatorial. Eventually, analytical considerations compelled the incorporationofcontinuous variables into the theory. This culminated in modern probability theory,the foundations of which were laid by Andrey Nikolaevich Kolmogorov. Kolmogorovcombined the notion ofsample space, introduced byRichard von Mises, andmeasure

theory and presented his axiom system for probability theory in 1933. Fairly quickly thisbecame the undisputed axiomatic basisfor modern probability theory.[2]

[edit] Treatment

Most introductions to probability theory treat discrete probability distributions andcontinuous probability distributions separately. The more mathematically advancedmeasure theory based treatment of probability covers both the discrete, the continuous,any mix of these two and more.

[edit] Discrete probability distributions

Main article:Discrete probability distribution

Discrete probability theory deals with events that occur in countable sample spaces.

Examples: Throwing dice, experiments with decks of cards, and random walk.
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Classical definition: Initially the probability of an event to occur was defined as numberof cases favorable for the event, over the number of total outcomes possible in anequiprobable sample space.

For example, if the event is "occurrence of an even number when a die is rolled", the

probability is given by , since 3 faces out of the 6 have even numbers and eachface has the same probability of appearing.

Modern definition: The modern definition starts with aset called the sample space,which relates to the set of allpossible outcomes in classical sense, denoted by

. It is then assumed that for each element , an intrinsic

"probability" value is attached, which satisfies the following properties:

1.

2.

That is, the probability functionf(x) lies between zero and one for every value ofx in thesample space , and the sum off(x) over all valuesx in the sample space is exactlyequal to 1. An event is defined as any subset of the sample space . The probabilityof the event defined as

So, the probability of the entire sample space is 1, and the probability of the null event is0.

The function mapping a point in the sample space to the "probability" value iscalled a probability mass function abbreviated as pmf. The modern definition does nottry to answer how probability mass functions are obtained; instead it builds a theory thatassumes their existence.

[edit] Continuous probability distributions

Main article: Continuous probability distribution

Continuous probability theory deals with events that occur in a continuous samplespace.

Classical definition: The classical definition breaks down when confronted with thecontinuous case. See Bertrand's paradox.

Modern definition: If the sample space is the real numbers ( ), then a function calledthe cumulative distribution function (orcdf) is assumed to exist, which gives
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for arandom variableX. That is,F(x) returns the probability thatXwill be less than or equal tox.

The cdf must satisfy the following properties.

1. is amonotonically non-decreasing,right-continuous function;

2.

3.

If is differentiable, then the random variableXis said to have a probability density

function orpdfor simply density

For a set , the probability of the random variableXbeing in is defined as

In case the probability density function exists, this can be written as

Whereas thepdfexists only for continuous random variables, the cdfexists for all randomvariables (including discrete random variables) that take values on

These concepts can be generalized formultidimensional cases on and othercontinuous sample spaces.

[edit] Measure-theoretic probability theory

The raison d'tre of the measure-theoretic treatment of probability is that it unifies thediscrete and the continuous, and makes the difference a question of which measure isused. Furthermore, it covers distributions that are neither discrete nor continuous normixtures 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 value fromrandom normal distribution with probability 1/2. It can still be studied to some extent by

considering it to have a pdf of , where [x] is the Kronecker deltafunction.
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Other distributions may not even be a mix, for example, theCantor distribution has nopositive probability for any single point, neither does it have a density. The modernapproach to probability theory solves these problems using measure theory to define theprobability space:

Given any set , (also called sample space) and a-algebra on it, a measurePiscalled a probability measure if

1. is non-negative;

2.

If is a Borel -algebra then there is a unique probability measure on for any cdf, andvice versa. The measure corresponding to a cdf is said to be induced by the cdf. Thismeasure coincides with the pmf for discrete variables, and pdf for continuous variables,making the measure-theoretic approach free of fallacies.

Theprobability of a set in the -algebra is defined as

where the integration is with respect to the measure induced by

Correlation Coefficient

A correlation coefficient is a number between -1 and 1 which measures the

degree to which two variables are linearly related. If there is perfect linear

relationship with positive slope between the two variables, we have a correlation

coefficient of 1; if there is positive correlation, whenever one variable has a high

(low) value, so does the other. If there is a perfect linear relationship with

negative slope between the two variables, we have a correlation coefficient of -1;

if there is negative correlation, whenever one variable has a high (low) value, the

other has a low (high) value. A correlation coefficient of 0 means that there is no

linear relationship between the variables.

There are a number of different correlation coefficients that might be appropriate

depending on the kinds of variables being studied.

See also Pearson's Product Moment Correlation Coefficient.

See also Spearman Rank Correlation Coefficient.
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Pearson's Product Moment Correlation Coefficient

Pearson's product moment correlation coefficient, usually denoted by r, is one

example of a correlation coefficient. It is a measure of the linear association

between two variables that have been measured on interval or ratio scales, such

as the relationship between height in inches and weight in pounds. However, it

can be misleadingly small when there is a relationship between the variables but

it is a non-linear one.

There are procedures, based on r, for making inferences about the population

correlation coefficient. However, these make the implicit assumption that the two

variables are jointly normally distributed. When this assumption is not justified, a

non-parametric measure such as the Spearman Rank Correlation Coefficient

might be more appropriate.

See also correlation coefficient.

Spearman Rank Correlation Coefficient

The Spearman rank correlation coefficient is one example of a correlation

coefficient. It is usually calculated on occasions when it is not convenient,

economic, or even possible to give actual values to variables, but only to assign a

rank order to instances of each variable. It may also be a better indicator that a

relationship exists between two variables when the relationship is non-linear.

Commonly used procedures, based on the Pearson's Product Moment

Correlation Coefficient, for making inferences about the population correlation

coefficient make the implicit assumption that the two variables are jointly normally

distributed. When this assumption is not justified, a non-parametric measure such

as the Spearman Rank Correlation Coefficient might be more appropriate.

See also correlation coefficient.
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Regression Equation

A regression equation allows us to express the relationship between two (or

more) variables algebraically. It indicates the nature of the relationship between

two (or more) variables. In particular, it indicates the extent to which you can

predict some variables by knowing others, or the extent to which some are

associated with others.

A linear regression equation is usually written

Y = a + bX + e

where

Y is the dependent variable

a is the intercept

b is the slope or regression coefficient

X is the independent variable (or covariate)

e is the error term

The equation will specify the average magnitude of the expected change in Y

given a change in X.

The regression equation is often represented on a scatterplot by a regression

line.

Regression Line

A regression line is a line drawn through the points on a scatterplot to summarisethe relationship between the variables being studied. When it slopes down (from

top left to bottom right), this indicates a negative or inverse relationship between

the variables; when it slopes up (from bottom right to top left), a positive or direct

relationship is indicated.

The regression line often represents the regression equation on a scatterplot.
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Simple Linear Regression

Simple linear regression aims to find a linear relationship between a response

variable and a possible predictor variable by the method of least squares.

Multiple Regression

Multiple linear regression aims is to find a linear relationship between a response

variable and several possible predictor variables.

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