chap05

Chap 5-*Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chapter 5

Discrete Random Variables and Probability DistributionsStatistics for Business and Economics 6th Edition

Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.

Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Chapter GoalsAfter completing this chapter, you should be able to: Interpret the mean and standard deviation for a discrete random variableUse the binomial probability distribution to find probabilitiesDescribe when to apply the binomial distributionUse the hypergeometric and Poisson discrete probability distributions to find probabilitiesExplain covariance and correlation for jointly distributed discrete random variables


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Introduction to Probability DistributionsRandom VariableRepresents a possible numerical value from a random experiment

Random VariablesDiscrete Random VariableContinuousRandom VariableCh. 5Ch. 6


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Discrete Random VariablesCan only take on a countable number of values

Examples:

Roll a die twiceLet X be the number of times 4 comes up (then X could be 0, 1, or 2 times)

Toss a coin 5 times. Let X be the number of heads (then X = 0, 1, 2, 3, 4, or 5)


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Experiment: Toss 2 Coins. Let X = # heads.TTDiscrete Probability Distribution4 possible outcomesTTHHHHProbability Distribution 0 1 2 x x Value Probability 0 1/4 = .25 1 2/4 = .50 2 1/4 = .25.50.25 Probability Show P(x) , i.e., P(X = x) , for all values of x:


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*P(x) 0 for any value of x

The individual probabilities sum to 1;

(The notation indicates summation over all possible x values)Probability DistributionRequired Properties


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Cumulative Probability FunctionThe cumulative probability function, denoted F(x0), shows the probability that X is less than or equal to x0

In other words,


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Expected Value Expected Value (or mean) of a discrete distribution (Weighted Average)

Example: Toss 2 coins, x = # of heads, compute expected value of x: E(x) = (0 x .25) + (1 x .50) + (2 x .25) = 1.0 x P(x) 0 .25 1 .50 2 .25


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Variance and Standard DeviationVariance of a discrete random variable X

Standard Deviation of a discrete random variable X


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Standard Deviation ExampleExample: Toss 2 coins, X = # heads, compute standard deviation (recall E(x) = 1)Possible number of heads = 0, 1, or 2


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Functions of Random VariablesIf P(x) is the probability function of a discrete random variable X , and g(X) is some function of X , then the expected value of function g is


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Linear Functions of Random VariablesLet a and b be any constants.

a)

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

b)

i.e., the expected value of bX is bE(x)


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Linear Functions of Random VariablesLet random variable X have mean x and variance 2x Let a and b be any constants. Let Y = a + bXThen the mean and variance of Y are

so that the standard deviation of Y is

(continued)


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Probability DistributionsContinuous Probability DistributionsBinomialHypergeometricPoissonProbability DistributionsDiscrete Probability DistributionsUniformNormalExponentialCh. 5Ch. 6


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*The Binomial DistributionBinomialHypergeometricPoissonProbability DistributionsDiscrete Probability Distributions


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Bernoulli DistributionConsider only two outcomes: success or failure Let P denote the probability of successLet 1 P be the probability of failure Define random variable X: x = 1 if success, x = 0 if failureThen the Bernoulli probability function is


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Bernoulli DistributionMean and VarianceThe mean is = P

The variance is 2 = P(1 P)


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Sequences of x Successes in n TrialsThe number of sequences with x successes in n independent trials is:

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

These sequences are mutually exclusive, since no two can occur at the same time


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Binomial Probability DistributionA fixed number of observations, ne.g., 15 tosses of a coin; ten light bulbs taken from a warehouseTwo mutually exclusive and collectively exhaustive categoriese.g., head or tail in each toss of a coin; defective or not defective light bulbGenerally called success and failureProbability of success is P , probability of failure is 1 PConstant probability for each observatione.g., Probability of getting a tail is the same each time we toss the coinObservations are independentThe outcome of one observation does not affect the outcome of the other


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Possible Binomial Distribution SettingsA manufacturing plant labels items as either defective or acceptableA firm bidding for contracts will either get a contract or notA marketing research firm receives survey responses of yes I will buy or no I will notNew job applicants either accept the offer or reject it


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*P(x) = probability of x successes in n trials, with probability of success P on each trial

x = number of successes in sample, (x = 0, 1, 2, ..., n) n = sample size (number of trials or observations) P = probability of success P(x)nx !nxP(1- P)XnX!() !=--Example: Flip a coin four times, let x = # heads:n = 4P = 0.51 - P = (1 - 0.5) = 0.5x = 0, 1, 2, 3, 4Binomial Distribution Formula


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Example: Calculating a Binomial ProbabilityWhat is the probability of one success in five observations if the probability of success is 0.1? x = 1, n = 5, and P = 0.1


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*n = 5 P = 0.1n = 5 P = 0.5Mean 0.2.4.6012345xP(x).2.4.6012345xP(x)0Binomial DistributionThe shape of the binomial distribution depends on the values of P and nHere, n = 5 and P = 0.1Here, n = 5 and P = 0.5


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Binomial DistributionMean and VarianceMean

Variance and Standard DeviationWheren = sample sizeP = probability of success(1 P) = probability of failure


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*n = 5 P = 0.1n = 5 P = 0.5Mean 0.2.4.6012345xP(x).2.4.6012345xP(x)0Binomial CharacteristicsExamples


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Using Binomial TablesExamples: n = 10, x = 3, P = 0.35: P(x = 3|n =10, p = 0.35) = .2522n = 10, x = 8, P = 0.45: P(x = 8|n =10, p = 0.45) = .0229

Nx p=.20p=.25p=.30p=.35p=.40p=.45p=.5010012345678910 0.10740.26840.30200.20130.08810.02640.00550.00080.00010.00000.00000.05630.18770.28160.25030.14600.05840.01620.00310.00040.00000.00000.02820.12110.23350.26680.20010.10290.03680.00900.00140.00010.00000.01350.07250.17570.25220.23770.15360.06890.02120.00430.00050.00000.00600.04030.12090.21500.25080.20070.11150.04250.01060.00160.00010.00250.02070.07630.16650.23840.23400.15960.07460.02290.00420.00030.00100.00980.04390.11720.20510.24610.20510.11720.04390.00980.0010


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Using PHStatSelect PHStat / Probability & Prob. Distributions / Binomial


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Using PHStatEnter desired values in dialog box

Here:n = 10p = .35

Output for x = 0 to x = 10 will be generated by PHStat

Optional check boxesfor additional output(continued)


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*P(x = 3 | n = 10, P = .35) = .2522PHStat OutputP(x > 5 | n = 10, P = .35) = .0949


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*The Hypergeometric DistributionBinomialPoissonProbability DistributionsDiscrete Probability DistributionsHypergeometric


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*The Hypergeometric Distributionn trials in a sample taken from a finite population of size NSample taken without replacementOutcomes of trials are dependentConcerned with finding the probability of X successes in the sample where there are S successes in the population


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Hypergeometric Distribution FormulaWhereN = population sizeS = number of successes in the population N S = number of failures in the populationn = sample sizex = number of successes in the sample n x = number of failures in the sample


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Using the Hypergeometric DistributionExample: 3 different computers are checked from 10 in the department. 4 of the 10 computers have illegal software loaded. What is the probability that 2 of the 3 selected computers have illegal software loaded?

N = 10n = 3 S = 4 x = 2The probability that 2 of the 3 selected computers have illegal software loaded is 0.30, or 30%.


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Hypergeometric Distribution in PHStatSelect:PHStat / Probability & Prob. Distributions / Hypergeometric


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Hypergeometric Distribution in PHStatComplete dialog box entries and get output N = 10 n = 3S = 4 x = 2 P(X = 2) = 0.3(continued)


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*The Poisson DistributionBinomialHypergeometricPoissonProbability DistributionsDiscrete Probability Distributions


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*The Poisson DistributionApply the Poisson Distribution when:You wish to count the number of times an event occurs in a given continuous intervalThe probability that an event occurs in one subinterval is very small and is the same for all subintervals The number of events that occur in one subinterval is independent of the number of events that occur in the other subintervalsThere can be no more than one occurrence in each subintervalThe average number of events per unit is (lambda)


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Poisson Distribution Formulawhere:x = number of successes per unit = expected number of successes per unite = base of the natural logarithm system (2.71828...)


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Poisson Distribution CharacteristicsMean

Variance and Standard Deviationwhere = expected number of successes per unit


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Using Poisson TablesExample: Find P(X = 2) if = .50

X0.100.200.300.400.500.600.700.800.90012345670.90480.09050.00450.00020.00000.00000.00000.00000.81870.16370.01640.00110.00010.00000.00000.00000.74080.22220.03330.00330.00030.00000.00000.00000.67030.26810.05360.00720.00070.00010.00000.00000.60650.30330.07580.01260.00160.00020.00000.00000.54880.32930.09880.01980.00300.00040.00000.00000.49660.34760.12170.02840.00500.00070.00010.00000.44930.35950.14380.03830.00770.00120.00020.00000.40660.36590.16470.04940.01110.00200.00030.0000


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Graph of Poisson ProbabilitiesP(X = 2) = .0758 Graphically: = .50

X =0.50012345670.60650.30330.07580.01260.00160.00020.00000.0000


Chart2

0.6065306597

0.3032653299

0.0758163325

0.0126360554

0.0015795069

0.0001579507

0.0000131626

0.0000009402

x

P(x)

Histogram

0

0.6065306597

0.3032653299

0.0758163325

0.0126360554

0.0015795069

0.0001579507

0.0000131626

0.0000009402

0.0000000588

0.0000000033

0.0000000002

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Number of Successes

P(X)

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Poisson2

Poisson Probabilities for Customer Arrivals

Data

Average/Expected number of successes:0.5

Poisson Probabilities Table

XP(X)P(=X)

00.6065310.6065310.0000000.3934691.000000

10.3032650.9097960.6065310.0902040.393469

20.0758160.9856120.9097960.0143880.090204

30.0126360.9982480.9856120.0017520.014388

40.0015800.9998280.9982480.0001720.001752

50.0001580.9999860.9998280.0000140.000172

60.0000130.9999990.9999860.0000010.000014

70.0000011.0000000.9999990.0000000.000001

80.0000001.0000001.0000000.0000000.000000

90.0000001.0000001.0000000.0000000.000000

100.0000001.0000001.0000000.0000000.000000

110.0000001.0000001.0000000.0000000.000000

120.0000001.0000001.0000000.0000000.000000

130.0000001.0000001.0000000.0000000.000000

140.0000001.0000001.0000000.0000000.000000

150.0000001.0000001.0000000.0000000.000000

160.0000001.0000001.0000000.0000000.000000

170.0000001.0000001.0000000.0000000.000000

180.0000001.0000001.0000000.0000000.000000

190.0000001.0000001.0000000.0000000.000000

200.0000001.0000001.0000000.0000000.000000

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XP(X)P(=X)

00.90480.9048370.0000000.0951631.000000

10.09050.9953210.9048370.0046790.095163

20.00450.9998450.9953210.0001550.004679

30.00020.9999960.9998450.0000040.000155

40.00001.0000000.9999960.0000000.000004

50.00001.0000001.0000000.0000000.000000

60.00001.0000001.0000000.0000000.000000

70.00001.0000001.0000000.0000000.000000

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Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Poisson Distribution ShapeThe shape of the Poisson Distribution depends on the parameter : = 0.50 = 3.00


Chart3

0.0497870684

0.1493612051

0.2240418077

0.2240418077

0.1680313557

0.1008188134

0.0504094067

0.0216040315

0.0081015118

0.0027005039

0.0008101512

0.0002209503

x

P(x)

Histogram

0

0.6065306597

0.3032653299

0.0758163325

0.0126360554

0.0015795069

0.0001579507

0.0000131626

0.0000009402

0.0000000588

0.0000000033

0.0000000002

0

0

0

0

0

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Number of Successes

P(X)

Histogram

Poisson2


Data

Average/Expected number of successes:3


XP(X)P(=X)

00.0497870.0497870.0000000.9502131.000000

10.1493610.1991480.0497870.8008520.950213

20.2240420.4231900.1991480.5768100.800852

30.2240420.6472320.4231900.3527680.576810

40.1680310.8152630.6472320.1847370.352768

50.1008190.9160820.8152630.0839180.184737

60.0504090.9664910.9160820.0335090.083918

70.0216040.9880950.9664910.0119050.033509

80.0081020.9961970.9880950.0038030.011905

90.0027010.9988980.9961970.0011020.003803

100.0008100.9997080.9988980.0002920.001102

110.0002210.9999290.9997080.0000710.000292

120.0000550.9999840.9999290.0000160.000071

130.0000130.9999970.9999840.0000030.000016

140.0000030.9999990.9999970.0000010.000003

150.0000011.0000000.9999990.0000000.000001

160.0000001.0000001.0000000.0000000.000000

170.0000001.0000001.0000000.0000000.000000

180.0000001.0000001.0000000.0000000.000000

190.0000001.0000001.0000000.0000000.000000

200.0000001.0000001.0000000.0000000.000000

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Data



XP(X)P(=X)

00.90480.9048370.0000000.0951631.000000

10.09050.9953210.9048370.0046790.095163

20.00450.9998450.9953210.0001550.004679

30.00020.9999960.9998450.0000040.000155

40.00001.0000000.9999960.0000000.000004

50.00001.0000001.0000000.0000000.000000

60.00001.0000001.0000000.0000000.000000

70.00001.0000001.0000000.0000000.000000

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Chart2

0.6065306597

0.3032653299

0.0758163325

0.0126360554

0.0015795069

0.0001579507

0.0000131626

0.0000009402

x

P(x)

Histogram

0

0.6065306597

0.3032653299

0.0758163325

0.0126360554

0.0015795069

0.0001579507

0.0000131626

0.0000009402

0.0000000588

0.0000000033

0.0000000002

0

0

0

0

0

0

0

0

0

0

Number of Successes

P(X)

Histogram

Poisson2


Data



XP(X)P(=X)

00.6065310.6065310.0000000.3934691.000000

10.3032650.9097960.6065310.0902040.393469

20.0758160.9856120.9097960.0143880.090204

30.0126360.9982480.9856120.0017520.014388

40.0015800.9998280.9982480.0001720.001752

50.0001580.9999860.9998280.0000140.000172

60.0000130.9999990.9999860.0000010.000014

70.0000011.0000000.9999990.0000000.000001

80.0000001.0000001.0000000.0000000.000000

90.0000001.0000001.0000000.0000000.000000

100.0000001.0000001.0000000.0000000.000000

110.0000001.0000001.0000000.0000000.000000

120.0000001.0000001.0000000.0000000.000000

130.0000001.0000001.0000000.0000000.000000

140.0000001.0000001.0000000.0000000.000000

150.0000001.0000001.0000000.0000000.000000

160.0000001.0000001.0000000.0000000.000000

170.0000001.0000001.0000000.0000000.000000

180.0000001.0000001.0000000.0000000.000000

190.0000001.0000001.0000000.0000000.000000

200.0000001.0000001.0000000.0000000.000000

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Poisson2

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P(x)

Poisson


Data



XP(X)P(=X)

00.90480.9048370.0000000.0951631.000000

10.09050.9953210.9048370.0046790.095163

20.00450.9998450.9953210.0001550.004679

30.00020.9999960.9998450.0000040.000155

40.00001.0000000.9999960.0000000.000004

50.00001.0000001.0000000.0000000.000000

60.00001.0000001.0000000.0000000.000000

70.00001.0000001.0000000.0000000.000000

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Sheet2

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Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Poisson Distribution in PHStatSelect:PHStat / Probability & Prob. Distributions / Poisson


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Poisson Distribution in PHStatComplete dialog box entries and get output P(X = 2) = 0.0758(continued)


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Joint Probability FunctionsA joint probability function is used to express the probability that X takes the specific value x and simultaneously Y takes the value y, as a function of x and y

The marginal probabilities are


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Conditional Probability FunctionsThe conditional probability function of the random variable Y expresses the probability that Y takes the value y when the value x is specified for X.

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


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*IndependenceThe jointly distributed random variables X and Y are said to be independent if and only if their joint probability function is the product of their marginal probability functions:

for all possible pairs of values x and y

A set of k random variables are independent if and only if


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*CovarianceLet X and Y be discrete random variables with means X and Y

The expected value of (X - X)(Y - Y) is called the covariance between X and Y

For discrete random variables

An equivalent expression is


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Covariance and IndependenceThe covariance measures the strength of the linear relationship between two variables

If two random variables are statistically independent, the covariance between them is 0 The converse is not necessarily true


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*CorrelationThe correlation between X and Y is:

= 0 no linear relationship between X and Y > 0 positive linear relationship between X and Ywhen X is high (low) then Y is likely to be high (low) = +1 perfect positive linear dependency < 0 negative linear relationship between X and Ywhen X is high (low) then Y is likely to be low (high) = -1 perfect negative linear dependency


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Portfolio AnalysisLet random variable X be the price for stock A Let random variable Y be the price for stock BThe market value, W, for the portfolio is given by the linear function

(a is the number of shares of stock A, b is the number of shares of stock B)


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Portfolio AnalysisThe mean value for W is

The variance for W is

or using the correlation formula

(continued)


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Example: Investment ReturnsReturn per $1,000 for two types of investmentsP(xiyi) Economic condition Passive Fund X Aggressive Fund Y .2 Recession- $ 25 - $200 .5 Stable Economy+ 50 + 60 .3 Expanding Economy + 100 + 350 InvestmentE(x) = x = (-25)(.2) +(50)(.5) + (100)(.3) = 50E(y) = y = (-200)(.2) +(60)(.5) + (350)(.3) = 95


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Computing the Standard Deviation for Investment ReturnsP(xiyi) Economic condition Passive Fund X Aggressive Fund Y 0.2 Recession- $ 25 - $200 0.5 Stable Economy+ 50 + 60 0.3 Expanding Economy + 100 + 350 Investment


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Covariance for Investment ReturnsP(xiyi) Economic condition Passive Fund X Aggressive Fund Y .2 Recession- $ 25 - $200 .5 Stable Economy+ 50 + 60 .3 Expanding Economy + 100 + 350 Investment


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Portfolio Example Investment X: x = 50 x = 43.30 Investment Y: y = 95 y = 193.21 xy = 8250

Suppose 40% of the portfolio (P) is in Investment X and 60% is in Investment Y:

The portfolio return and portfolio variability are between the values for investments X and Y considered individually


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Interpreting the Results for Investment ReturnsThe aggressive fund has a higher expected return, but much more risk

y = 95 > x = 50 buty = 193.21 > x = 43.30

The Covariance of 8250 indicates that the two investments are positively related and will vary in the same direction


Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 5-*Chapter SummaryDefined discrete random variables and probability distributionsDiscussed the Binomial distributionDiscussed the Hypergeometric distributionReviewed the Poisson distributionDefined covariance and the correlation between two random variablesExamined application to portfolio investment


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