Chapter 5Chapter 5

Discrete Random Discrete Random Variables and Variables and

Probability Probability DistributionsDistributions

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Random VariablesRandom Variables

A random variablerandom variable is a variable that takes on numerical values determined by the outcome of a random experiment.

Discrete Random VariablesDiscrete Random Variables

A random variable is discrete discrete if it can take on no more than a countable number of values.

Discrete Random VariablesDiscrete Random Variables(Examples)(Examples)

1. The number of defective items in a sample of twenty items taken from a large shipment.

2. The number of customers arriving at a check-out counter in an hour.

3. The number of errors detected in a corporation’s accounts.

4. The number of claims on a medical insurance policy in a particular year.

Continuous Random Continuous Random VariablesVariables

A random variable is continuous continuous if it can take any value in an interval.

Continuous Random Continuous Random VariablesVariables

(Examples)(Examples)

1. The income in a year for a family.2. The amount of oil imported into the U.S.

in a particular month.3. The change in the price of a share of IBM

common stock in a month.4. The time that elapses between the

installation of a new computer and its failure.

5. The percentage of impurity in a batch of chemicals.

Discrete Probability Discrete Probability DistributionsDistributions

The probability distribution function probability distribution function (DPF),(DPF), P(x), of a discrete random variable expresses the probability that X takes the value x, as a function of x. That is

. of valuesallfor ),()( xxXPxP

Discrete Probability Discrete Probability DistributionsDistributions

(Example 5.1)(Example 5.1)

Graph the probability distribution function for the roll of a single six-sided die.

1 2 3 4 5 6

1/6

P(x)

x

Figure 5.1Figure 5.1

Required Properties of Required Properties of Probability Distribution Probability Distribution

Functions of Discrete Random Functions of Discrete Random VariablesVariables

Let X be a discrete random variable with probability distribution function, P(x). Then

i. P(x) 0 for any value of xii. The individual probabilities sum to 1; that is

Where the notation indicates summation over all possible values x.

x

xP 1)(

Cumulative Probability Cumulative Probability FunctionFunction

The cumulative probability function,cumulative probability function, F(x0), of a random variable X expresses the probability that X does not exceed the value x0, as a function of x0. That is

Where the function is evaluated at all values x0

)()( 00 xXPxF

Derived Relationship Between Derived Relationship Between Probability Function and Cumulative Probability Function and Cumulative

Probability FunctionProbability Function

Let X be a random variable with probability function P(x) and cumulative probability function F(x0). Then it can be shown that

Where the notation implies that summation is over all possible values x that are less than or equal to x0.

0

)()( 0xx

XPxF

Derived Properties of Derived Properties of Cumulative Probability Cumulative Probability

Functions for Discrete Random Functions for Discrete Random VariablesVariables

Let X be a discrete random variable with a cumulative probability function, F(x0). Then we can show that

i. 0 F(x0) 1 for every number x0

ii. If x0 and x1 are two numbers with x0 < x1, then F(x0) F(x1)

Expected ValueExpected Value

The expected value, E(X),expected value, E(X), of a discrete random variable X is defined

Where the notation indicates that summation extends over all possible values x.The expected value of a random variable is called its meanmean and is denoted xx.

x

xxPXE )()(

Expected Value: Functions Expected Value: Functions of Random Variablesof Random Variables

Let X be a discrete random variable with probability function P(x) and let g(X) be some function of X. Then the expected value, E[g(X)], of that function is defined as

x

xPxgXgE )()()]([

Variance and Standard Variance and Standard DeviationDeviation

Let X be a discrete random variable. The expectation of the squared discrepancies about the mean, (X - )2, is called the variancevariance, denoted 2

x and is given by

The standard deviationstandard deviation, x , is the positive square root of the variance.

x

xxx xPxXE )()()( 222

VarianceVariance(Alternative Formula)(Alternative Formula)

The variancevariance of a discrete random variable X can be expressed as

22

222

)(

)(

xx

xx

xPx

XE

Expected Value and Variance for Expected Value and Variance for Discrete Random Variable Using Discrete Random Variable Using

Microsoft ExcelMicrosoft Excel(Figure 5.4)(Figure 5.4)

Sales P(x) Mean Variance0 0.15 0 0.5703751 0.3 0.3 0.270752 0.2 0.4 0.00053 0.2 0.6 0.22054 0.1 0.4 0.420255 0.05 0.25 0.465125

1.95 1.9475

Expected Value = 1.95 Variance = 1.9475

Summary of Properties for Summary of Properties for Linear Function of a Random Linear Function of a Random

VariableVariable

Let X be a random variable with mean x , and variance 2

x ; and let a and b be any constant fixed numbers. Define the random variable Y = a + bX. Then, the meanmean and variancevariance of Y are

and

so that the standard deviation of Ystandard deviation of Y is

XY babXaE )(

XY bbXaVar 222 )(

XY b

Summary Results for the Mean Summary Results for the Mean and Variance of Special Linear and Variance of Special Linear

FunctionsFunctions

a) Let b = 0 in the linear function, W = a + bX. Then W = a (for any constant a).

If a random variable always takes the value a, it will have a mean a and a variance 0.

b) Let a = 0 in the linear function, W = a + bX. Then W = bX.

0)()( aVarandaaE

22)()( XX baVarandbbXE

Mean and Variance of ZMean and Variance of Z

Let a = -X/X and b = 1/ X in the linear function Z = a + bX. Then,

so that

and

X

XXbXaZ

01

X

XX

X

X

XXE

11 2

2

X

XX

XXVar

Bernoulli DistributionBernoulli Distribution

A Bernoulli distributionBernoulli distribution arises from a random experiment which can give rise to just two possible outcomes. These outcomes are usually labeled as either “success” or “failure.” If denotes the probability of a success and the probability of a failure is (1 - ), the the Bernoulli probability function is )1()1()0( PandP

Mean and Variance of a Mean and Variance of a Bernoulli Random VariableBernoulli Random Variable

The meanmean is:

And the variancevariance is:

)1()1)(0()()( xPxXEX

X

)1()1()1()0(

)()(])[(

22

222

X

XXX xPxXE

Sequences of Sequences of xx Successes in Successes in nn Trials Trials

The number of sequences with x successes number of sequences with x successes in n independent trialsin n independent trials is:

Where n! = n x (x – 1) x (n – 2) x . . . x 1 and 0! = 1.

)!(!

!

xnx

nC n

x

time. samethe at occur can them of two no since

exclusive,mutually are sequencesC These nx

Binomial DistributionBinomial Distribution

Suppose that a random experiment can result in two possible mutually exclusive and collectively exhaustive outcomes, “success” and “failure,” and that is the probability of a success resulting in a single trial. If n independent trials are carried out, the distribution of the resulting number of successes “x” is called the binomial binomial distributiondistribution. Its probability distribution function for the binomial random variable X = x is:

P(x successes in n independent trials)=

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

)()1()!(!

!)( xnx

xnx

nxP

Mean and Variance of a Mean and Variance of a Binomial Probability Binomial Probability

DistributionDistributionLet X be the number of successes in n independent trials, each with probability of success . The x follows a binomial distribution with meanmean,

and variancevariance,

nXEX )(

)1(])[( 22 nXEX

Binomial ProbabilitiesBinomial Probabilities- An Example –- An Example –

(Example 5.7)(Example 5.7)

An insurance broker, Shirley Ferguson, has five contracts, and she believes that for each contract, the probability of making a sale is 0.40.

What is the probability that she makes at most one sale?

P(at most one sale) = P(X 1) = P(X = 0) + P(X = 1)

= 0.078 + 0.259 = 0.337

259.0)6.0()4.0(1!4!

5! P(1) sale) P(1

0.078 )6.0()4.0(0!5!

5! P(0) sales) P(no

41

50

Binomial Probabilities, n = 100, Binomial Probabilities, n = 100, =0.40=0.40

(Figure 5.10)(Figure 5.10)

Sample size 100Probability of success 0.4Mean 40Variance 24Standard deviation 4.898979

Binomial Probabilities TableX P(X) P(<=X) P(<X) P(>X) P(>=X)

36 0.059141 0.238611 0.179469 0.761389 0.82053137 0.068199 0.30681 0.238611 0.69319 0.76138938 0.075378 0.382188 0.30681 0.617812 0.6931939 0.079888 0.462075 0.382188 0.537925 0.61781240 0.081219 0.543294 0.462075 0.456706 0.53792541 0.079238 0.622533 0.543294 0.377467 0.45670642 0.074207 0.69674 0.622533 0.30326 0.37746743 0.066729 0.763469 0.69674 0.236531 0.30326

Hypergeometric DistributionHypergeometric DistributionSuppose that a random sample of n objects is chosen from a group of N objects, S of which are successes. The distribution of the number of X successes in the sample is called the hypergeometric distributionhypergeometric distribution. Its probability function is:

Where x can take integer values ranging from the larger of 0 and [n-(N-S)] to the smaller of n and S.

)!(!!

)!()!()!(

)!(!!

)(

nNnN

xnSNxnSN

xSxS

C

CCxP

Nn

SNxn

Sx

Poisson Probability Poisson Probability DistributionDistribution

Assume that an interval is divided into a very large number of subintervals so that the probability of the occurrence of an event in any subinterval is very small. The assumptions of a Poisson Poisson probability distributionprobability distribution are:

1) The probability of an occurrence of an event is constant for all subintervals.

2) There can be no more than one occurrence in each subinterval.

3) Occurrences are independent; that is, the number of occurrences in any non-overlapping intervals in independent of one another.

Poisson Probability Poisson Probability DistributionDistribution

The random variable X is said to follow the Poisson probability distribution if it has the probability function:

whereP(x) = the probability of x successes over a given

period of time or space, given = the expected number of successes per time or

space unit; > 0e = 2.71828 (the base for natural logarithms)

The mean and variance of the Poisson probability mean and variance of the Poisson probability distribution aredistribution are:

1,2,... 0,xfor,!

)(

x

exP

x

])[()( 22 XEandXE xx

Partial Poisson Probabilities for Partial Poisson Probabilities for = = 0.03 Obtained Using Microsoft 0.03 Obtained Using Microsoft

Excel Excel PPHStatHStat(Figure 5.14)(Figure 5.14)

Poisson Probabilities TableX P(X) P(<=X) P(<X) P(>X) P(>=X)0 0.970446 0.970446 0.000000 0.029554 1.0000001 0.029113 0.999559 0.970446 0.000441 0.0295542 0.000437 0.999996 0.999559 0.000004 0.0004413 0.000004 1.000000 0.999996 0.000000 0.0000044 0.000000 1.000000 1.000000 0.000000 0.000000

Poisson Approximation to Poisson Approximation to the Binomial Distributionthe Binomial Distribution

Let X be the number of successes resulting from n independent trials, each with a probability of success, . The distribution of the number of successes X is binomial, with mean n. If the number of trials n is large and n is of only moderate size (preferably n 7), this distribution can be approximated by the Poisson distribution with = n. The probability function of the approximating distribution is then:

1,2,... 0,xfor,!

)()(

x

nexP

xn

Joint Probability FunctionsJoint Probability Functions

Let X and Y be a pair of discrete random variables. Their joint probability functionjoint probability function expresses the probability that X takes the specific value x and simultaneously Y takes the value y, as a function of x and y. The notation used is P(x, y) so,

)(),( yYxXPyxP

Joint Probability FunctionsJoint Probability Functions

Let X and Y be a pair of jointly distributed random variables. In this context the probability function of the random variable X is called its marginal probability functionmarginal probability function and is obtained by summing the joint probabilities over all possible values; that is,

Similarly, the marginal probability functionmarginal probability function of the random variable Y is

y

yxPxP ),()(

x

yxPyP ),()(

Properties of Joint Properties of Joint Probability FunctionsProbability Functions

Let X and Y be discrete random variables with joint probability function P(x,y). Then

1.1. P(x,y) P(x,y) 0 for any pair of values x and y 0 for any pair of values x and y

2.2. The sum of the joint probabilities P(x, The sum of the joint probabilities P(x, y) over all possible values must be 1.y) over all possible values must be 1.

Conditional Probability Conditional Probability FunctionsFunctions

Let X and Y be a pair of jointly distributed discrete random variables. The conditional probability functionconditional probability function of the random variable Y, given that the random variable X takes the value x, expresses the probability that Y takes the value y, as a function of y, when the value x is specified for X. This is denoted P(y|x), and so by the definition of conditional probability:

Similarly, the conditional probability functionconditional probability function of X, given Y = y is:

)(

),()|(

xP

yxPxyP

)(

),()|(

yP

yxPyxP

Independence of Jointly Independence of Jointly Distributed Random Distributed Random

VariablesVariables

The jointly distributed random variables X and Y are said to be independentindependent if and only if their joint probability function is the product of their marginal probability functions, that is, if and only if

And k random variables are independent if and only if

y. and x valuesof pairs possible allfor )()(),( yPxPyxP

)()()(),,,( 2121 kk xPxPxPxxxP

Expected Value Function of Expected Value Function of Jointly Distributed Random Jointly Distributed Random

VariablesVariables

Let X and Y be a pair of discrete random variables with joint probability function P(x, y). The expectation of any function g(x, y)expectation of any function g(x, y) of these random variables is defined as:

x y

yxPyxgYXgE ),(),()],([

Stock Returns, Marginal Stock Returns, Marginal Probability, Mean, VarianceProbability, Mean, Variance

(Example 5.16)(Example 5.16)

Y Return

X Retur

n

0% 5% 10% 15%

0% 0.0625

0.0625

0.0625

0.0625

5% 0.0625

0.0625

0.0625

0.0625

10% 0.0625

0.0625

0.0625

0.0625

15% 0.0625

0.0625

0.0625

0.0625

Table 5.6

CovarianceCovariance

Let X be a random variable with mean X , and let Y be a random variable with mean, Y . The expected value of (X - X )(Y - Y ) is called the covariance covariance between X and Y, denoted Cov(X, Y).For discrete random variables

An equivalent expression is

x y

yxYX yxPyxYXEYXCov ),())(()])([(),(

x y

yxyx yxxyPXYEYXCov ),()(),(

CorrelationCorrelation

Let X and Y be jointly distributed random variables. The correlation between X and Y is:

YX

YXCovYXCorr

),(

),(

Covariance and Statistical Covariance and Statistical IndependenceIndependence

If two random variables are statistically statistically independentindependent, the covariance between them is 0. However, the converse is not necessarily true.

Portfolio AnalysisPortfolio Analysis

The random variable X is the price for stock A and the random variable Y is the price for stock B. The market value, W, for the portfolio is given by the linear function,

Where, a, is the number of shares of stock A and, b, is the number of shares of stock B.

bYaXW

Portfolio AnalysisPortfolio Analysis

The mean value for Wmean value for W is,

The variance for Wvariance for W is,

or using the correlation,

YX

W

ba

bYaXEWE

][][

),(222222 YXabCovba YXW

YXYXW YXabCorrba ),(222222

Key WordsKey Words Bernoulli Random

Variable, Mean and Variance

Binomial Distribution Conditional Probability

Function Continuous Random

Variable Correlation Covariance Cumulative Probability

Function

Differences of Random Variables

Discrete Random Variable Expected Value Expected Value: Functions

of Random Variables Expected Value: Function of

Jointly Distributed Random Variable

Hypergeometric Distribution Independence of Jointly

Distributed Random Variables

Key WordsKey Words(continued)(continued)

Joint Probability Function Marginal Probability

Function Mean of Binomial

Distribution Mean: Functions of

Random Variables Poisson Approximation

to the Binomial Distribution

Poisson Distribution Portfolio Analysis

Portfolio, Market Value Probability Distribution

Function Properties: Cumulative

Probability Functions Properties: Joint

Probability Functions Properties: Probability

Distribution Functions Random Variable

Key WordsKey Words(continued)(continued)

Relationships: Probability Function and Cumulative Probability Function

Standard Deviation: Discrete Random Variable

Sums of Random Variables

Variance: Binomial Distribution

Variance: Discrete Random Variable

Variance: Discrete Random Variable (Alternative Formula)

Variance: Functions of Random Variables