basic statistic chap05

Chapter 5

Some Important Discrete Probability DistributionsStatistics for ManagersUsing Microsoft Excel 4th Edition

Chapter GoalsAfter completing this chapter, you should be able to: Compute the mean and standard deviation for a discrete probability distributionUse the binomial, hypergeometric and Poisson discrete probability distributions to find probabilitiesDescribe when to apply the binomial, hypergeometric and Poisson distributions

Introduction to Probability DistributionsRandom VariableRepresents a possible numerical value from an uncertain event

Random VariablesDiscrete Random VariableContinuousRandom VariableCh. 5Ch. 6

Discrete Random VariablesCan only assume 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)

Discrete Probability DistributionX Value Probability 0 1/4 = .25 1 2/4 = .50 2 1/4 = .25Experiment: Toss 2 Coins. Let X = # heads.TT4 possible outcomesTTHHHHProbability Distribution 0 1 2 X .50.25 Probability

Discrete Random Variable Summary Measures Expected Value 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

Discrete Random Variable Summary MeasuresVariance of a discrete random variable

Standard Deviation of a discrete random variable

where:E(X) = Expected value of the discrete random variable X Xi = the ith outcome of XP(Xi) = Probability of the ith occurrence of X(continued)

Computing the Mean for 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

Computing the Standard Deviation 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

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

Probability DistributionsContinuous Probability DistributionsBinomialHypergeometricPoissonProbability DistributionsDiscrete Probability DistributionsNormalUniformExponentialCh. 5Ch. 6

The Binomial DistributionBinomialHypergeometricPoissonProbability DistributionsDiscrete Probability Distributions

Binomial Probability DistributionA fixed number of observations, ne.g.: 15 tosses of a coin; ten light bulbs taken from a shipmentTwo 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 coin

Binomial Probability Distribution(continued)Observations are independentThe outcome of one observation does not affect the outcome of the otherTwo sampling methodsInfinite population without replacementFinite population with replacement

ExamplesA 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 not buyNew job applicants either accept the offer or reject it

Rule of CombinationsThe number of combinations of selecting X objects out of n objects is where:n! =n(n - 1)(n - 2) . . . (2)(1)X! = X(X - 1)(X - 2) . . . (2)(1) 0! = 1 (by definition)

Binomial Distribution FormulaP(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 - .5) = .5X = 0, 1, 2, 3, 4

Example: Calculating a Binomial ProbabilityWhat is the probability of one success in four flips if the probability of success is .5? X = 1, n = 4, and p = .5

Binomial DistributionThe shape of the binomial distribution depends on the values of p and nn = 5 p = 0.1n = 5 p = 0.5Mean 0.2.4.6012345XP(X).2.4.6012345XP(X)0Here, n = 5 and p = .1Here, n = 5 and p = .5 (all distributions for p=.5 are symmetrical)

Binomial Distribution CharacteristicsMean

Variance and Standard DeviationWheren = sample sizep = probability of success(1 p) = probability of failure

Binomial Characteristicsn = 5 p = 0.1n = 5 p = 0.5Mean 0.2.4.6012345XP(X).2.4.6012345XP(X)0Examples

Using PHStatSelect PHStat / Probability & Prob. Distributions / Binomial

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)

PHStat OutputP(X = 3 | n = 10, p = .35) = .2522P(X > 5 | n = 10, p = .35) = .0949

The Hypergeometric DistributionBinomialPoissonProbability DistributionsDiscrete Probability DistributionsHypergeometric

The Hypergeometric Distributionn trials in a sample taken from a finite population of size NSample taken without replacementTrials are dependentConcerned with finding the probability of X successes in the sample where there are A successes in the population

Hypergeometric Distribution FormulaWhereN = Population sizeA = number of successes in the population N A = number of failures in the populationn = sample sizeX = number of successes in the sample n X = number of failures in the sample(Two possible outcomes per trial)

Properties of the Hypergeometric DistributionThe mean of the hypergeometric distribution is

The standard deviation is

Where is called the Finite Population Correction Factor from sampling without replacement from a finite population

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 A = 4 X = 2The probability that 2 of the 3 selected computers will have illegal software loaded is .30

Hypergeometric Distribution in PHStatSelect:PHStat / Probability & Prob. Distributions / Hypergeometric

Hypergeometric Distribution in PHStatComplete dialog box entries and get output N = 10 n = 3A = 4 X = 2 P(X = 2) = 0.3(continued)

The Poisson DistributionBinomialHypergeometricPoissonProbability DistributionsDiscrete Probability Distributions

The Poisson DistributionApply the Poisson Distribution when:You wish to count the number of times an event occurs in a given intervalThe probability that an event occurs in the interval is the same for all intervals of equal size The number of events that occur in one interval is independent of the number of events that occur in the other intervalsThe average number of events per interval or unit is (lambda)

Poisson Distribution Formulawhere:X = number of successes per unit = expected number of successes per intervale = base of the natural logarithm system (2.71828...)

Poisson Distribution CharacteristicsMean

Variance and Standard Deviationwhere = expected number of successes per unit

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

0

0

0

0

0

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Number of Successes

P(X)

Histogram

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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Poisson2

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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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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

0

0

0

0

0

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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Poisson2

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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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Sheet2

Sheet3

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)

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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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Sheet1

Sheet2

Sheet3

Poisson Distribution in PHStatSelect:PHStat / Probability & Prob. Distributions / Poisson

Poisson Distribution in PHStatComplete dialog box entries and get output P(X = 2) = 0.0758(continued)

Chapter SummaryAddressed the probability of a discrete random variableDiscussed the Binomial distributionDiscussed the Poisson distributionDiscussed the Hypergeometric distribution

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basic statistic chap05