二元随机变数之机率分配

1、Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-1Chapter 5Some Important Discrete Probability DistributionsBasic Business Statistics10th EditionBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-2Learning ObjectivesIn this chapter, you learn: The properties of a probability

2、distributionTo calculate the expected value and variance of a probability distributionTo calculate the covariance and its use in financeTo calculate probabilities from binomial, Poisson distributions and hypergeometric How to use the binomial, hypergeometric, and Poisson distributions to solve busin

3、ess problemsBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-3Introduction to Probability DistributionsRandom VariableRepresents a possible numerical value from an uncertain eventRandom VariablesDiscrete Random VariableContinuousRandom VariableCh. 5Ch. 6Basic Business Statistics, 10e 20

4、06 Prentice-Hall, Inc.Chap 5-4Discrete Random VariablesCan only assume a countable number of valuesExamples: 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)Basic Business Statis

5、tics, 10e 2006 Prentice-Hall, Inc.Chap 5-5Experiment: Toss 2 Coins. Let X = # heads.TTDiscrete Probability Distribution4 possible outcomesTTHHHHProbability Distribution 0 1 2 X X Value Probability 0 1/4 = 0.25 1 2/4 = 0.50 2 1/4 = 0.250.500.25 Probability Basic Business Statistics, 10e 2006 Prentice

6、-Hall, Inc.Chap 5-6Discrete Random Variable Summary Measures 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 0.25) + (1 x 0.50) + (2 x 0.25) = 1.0 X P(X) 0 0.25 1 0.50 2 0.25N1iii)X(PX E(X) Basic Bu

7、siness Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-7Variance of a discrete random variableStandard Deviation of a discrete random variablewhere:E(X) = Expected value of the discrete random variable X Xi = the ith outcome of XP(Xi) = Probability of the ith occurrence of XDiscrete Random Variable S

8、ummary MeasuresN1ii2i2)P(XE(X)X(continued)N1ii2i2)P(XE(X)XBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-8Example: Toss 2 coins, X = # heads, compute standard deviation (recall E(X) = 1)Discrete Random Variable Summary Measures)P(XE(X)Xi2i0.7070.50(0.25)1)(2(0.50)1)(1(0.25)1)(0222(con

9、tinued)Possible number of heads = 0, 1, or 2Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-9二元隨機變數之機率分配Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-10事件機率事件機率Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-11表表5.6 一般化的聯合機率分配表一般化的聯合機率分配表Basic Business Sta

10、tistics, 10e 2006 Prentice-Hall, Inc.Chap 5-12()() (,)(,)(,)(,)(,)()()( )( )ijijijiijjijijijiijjijijjiiijjijZABE ABAB P A BAP A BB P A BAP A BBP A BAP AB P BE AE B Expected ValueBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-13The CovarianceThe covariance measures the strength of the

11、linear relationship between two variablesThe covariance:)YX(P)Y(EY)(X(EXN1iiiiiXYwhere:X = discrete variable XXi = the ith outcome of XY = discrete variable YYi = the ith outcome of YP(XiYi) = probability of occurrence of the ith outcome of X and the ith outcome of YBasic Business Statistics, 10e 20

12、06 Prentice-Hall, Inc.Chap 5-14Computing the Mean for Investment ReturnsReturn per $1,000 for two types of investmentsP(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 InvestmentE(X) = X = (-25)(0.2) +

13、(50)(0.5) + (100)(0.3) = 50E(Y) = Y = (-200)(0.2) +(60)(0.5) + (350)(0.3) = 95Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-15Computing the Standard Deviation for Investment ReturnsP(XiYi) Economic condition Passive Fund X Aggressive Fund Y 0.2 Recession- $ 25 - $200 0.5 Stable Econo

14、my+ 50 + 60 0.3 Expanding Economy + 100 + 350 Investment43.30(0.3)50)(100(0.5)50)(50(0.2)50)(-25222X193.71(0.3)95)(350(0.5)95)(60(0.2)95)(-200222YBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-16Computing the Covariance for Investment ReturnsP(XiYi) Economic condition Passive Fund X A

15、ggressive Fund Y 0.2 Recession- $ 25 - $200 0.5 Stable Economy+ 50 + 60 0.3 Expanding Economy + 100 + 350 Investment825095)(0.3)50)(350(10095)(0.5)50)(60(5095)(0.2)200-50)(-25XYBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-17Interpreting the Results for Investment ReturnsThe aggressi

16、ve fund has a higher expected return, but much more risk Y = 95 X = 50 butY = 193.71 X = 43.30The Covariance of 8250 indicates that the two investments are positively related and will vary in the same directionBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-18The Sum of Two Random Vari

17、ablesExpected Value of the sum of two random variables:Variance of the sum of two random variables:Standard deviation of the sum of two random variables:XY2Y2X2YX2Y)Var(X)Y(E)X(EY)E(X2YXYXBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-19Portfolio Expected Return and Portfolio RiskPort

18、folio expected return (weighted average return):Portfolio risk (weighted variability)Where w = portion of portfolio value in asset X (1 - w) = portion of portfolio value in asset Y)Y(E)w1()X(EwE(P)XY2Y22X2Pw)-2w(1)w1(wBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-20Portfolio Example

19、Investment X: X = 50 X = 43.30 Investment Y: Y = 95 Y = 193.21 XY = 8250Suppose 40% of the portfolio 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 individually77(95)(0.6)(50)0.4E(P)133.30)(8250)2

20、(0.4)(0.6(193.71)(0.6)(43.30)(0.4)2222PBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-21Probability DistributionsContinuous Probability DistributionsBinomialHypergeometricPoissonProbability DistributionsDiscrete Probability DistributionsNormalUniformExponentialCh. 5Ch. 6Basic Business

21、 Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-22The Binomial DistributionBinomialHypergeometricPoissonProbability DistributionsDiscrete Probability DistributionsBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-23Binomial Probability DistributionA fixed number of observations, ne.g., 1

22、5 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 “failure”Probability of success is p, probability of failure is 1 pCon

23、stant probability for each observatione.g., Probability of getting a tail is the same each time we toss the coinBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-24Binomial Probability Distribution(continued)Observations are independentThe outcome of one observation does not affect the o

24、utcome of the otherTwo sampling methodsInfinite population without replacementFinite population with replacementBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-25Possible Binomial Distribution SettingsA manufacturing plant labels items as either defective or acceptableA firm bidding fo

25、r contracts will either get a contract or notA marketing research firm receives survey responses of “yes I will buy” or “no I will not”New job applicants either accept the offer or reject itBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-26Rule of CombinationsThe number of combinations

26、 of selecting X objects out of n objects isX)!(nX!n!Cxn where:n! =(n)(n - 1)(n - 2) . . . (2)(1)X! = (X)(X - 1)(X - 2) . . . (2)(1) 0! = 1 (by definition)Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-27P(X) = probability of X successes in n trials, with probability of success p on ea

27、ch 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

28、Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-28Example: Calculating a Binomial ProbabilityWhat is the probability of one success in five observations if the probability of success is .1? X = 1, n = 5, and p = 0.10.32805.9)(5)(0.1)(00.1)(1(0.1)1)!(51!5!p)(1pX)!(nX!n!1)P(X4151XnXBasic

29、 Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-29n = 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 = 0.5Basic Business Statistics, 10e 2006

30、Prentice-Hall, Inc.Chap 5-30Binomial Distribution CharacteristicsMeanVariance and Standard DeviationnpE(x)p)-np(12p)-np(1 Wheren = sample sizep = probability of success(1 p) = probability of failureBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-31Basic Business Statistics, 10e 2006 Pr

31、entice-Hall, Inc.Chap 5-32Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-33Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-34Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-35The Bernoulli DistributionSuppose that a random experiment can give rise to just t

32、wo possible mutually exclusive and collectively exhaustive outcomes, which for convenience we will label “success” and “failure.” Let p denote the probability of success, so that the probability of failure is (1-p). Now define the random variable X so that X takes the value 1 if the outcome of the e

33、xperiment is success and 0 otherwise.The probability function of this random variable is thenPx(0) = (1-p), Px(1) = p, This distribution is known as the Bernoulli distribution. Its mean and variance areE(X) = 0*(1-p)+1*(p) = pV(X) = (0-p)2(1-p)+(1-p)2p = p(1-p)Basic Business Statistics, 10e 2006 Pre

34、ntice-Hall, Inc.Chap 5-36Binomial DistributionExperiment involves n identical trialsEach trial has exactly two possible outcomes: success and failureEach trial is independent of the previous trialsp is the probability of a success on any one trialq = (1-p) is the probability of a failure on any one

35、trialp and q are constant throughout the experimentX is the number of successes in the n trialsApplicationsSampling with replacement Sampling without replacement - n 5% NBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-37n = 5 p = 0.1n = 5 p = 0.5Mean 0.2.4.6012345XP(X).2.4.6012345XP(X)

36、00.5(5)(0.1)np0.67080.1)(5)(0.1)(1p)np(1-2.5(5)(0.5)np1.1180.5)(5)(0.5)(1p)np(1-Binomial CharacteristicsExamplesBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-38Using Binomial Tablesn = 10 xp=.20p=.25p=.30p=.35p=.40p=.45p=.500123456789100.10740.26840.30200.20130.08810.02640.00550.0008

37、0.00010.00000.00000.05630.18770.28160.25030.14600.05840.01620.00310.00040.00000.00000.02820.12110.23350.26680.20010.10290.03680.00900.00140.00010.00000.01350.07250.17570.25220.23770.15360.06890.02120.00430.00050.00000.00600.04030.12090.21500.25080.20070.11150.04250.01060.00160.00010.00250.02070.0763

38、0.16650.23840.23400.15960.07460.02290.00420.00030.00100.00980.04390.11720.20510.24610.20510.11720.04390.00980.0010109876543210p=.80p=.75p=.70p=.65p=.60p=.55p=.50 xExamples: n = 10, p = 0.35, x = 3: P(x = 3|n =10, p = 0.35) = 0.2522n = 10, p = 0.75, x = 2: P(x = 2|n =10, p = 0.75) = 0.0004Basic Busin

39、ess Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-39The Hypergeometric DistributionBinomialPoissonProbability DistributionsDiscrete Probability DistributionsHypergeometricBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-40The Hypergeometric Distribution“n” trials in a sample taken from

40、 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 “A” successes in the populationBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-41Hypergeometric Distributi

41、on FormulanNXnANXACCCP(X)nNXnANXAWhereN = 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 sampleBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-42Prop

42、erties of the Hypergeometric DistributionThe mean of the hypergeometric distribution isThe standard deviation isWhere is called the “Finite Population Correction Factor” from sampling without replacement from a finite populationNnAE(x)1- Nn-NNA)-nA(N21- Nn-NBasic Business Statistics, 10e 2006 Prenti

43、ce-Hall, Inc.Chap 5-43Using the Hypergeometric Distribution Example: 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 = 20.3120

44、(6)(6)3101624nNXnANXA2)P(XThe probability that 2 of the 3 selected computers have illegal software loaded is 0.30, or 30%.Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-44The Poisson DistributionBinomialHypergeometricPoissonProbability DistributionsDiscrete Probability DistributionsBa

45、sic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-45The Poisson DistributionApply the Poisson Distribution when:You wish to count the number of times an event occurs in a given area of opportunityThe probability that an event occurs in one area of opportunity is the same for all areas of o

46、pportunity The number of events that occur in one area of opportunity is independent of the number of events that occur in the other areas of opportunityThe probability that two or more events occur in an area of opportunity approaches zero as the area of opportunity becomes smallerThe average numbe

47、r of events per unit is (lambda)Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-46Poisson Distribution Formulawhere:X = number of events in an area of opportunity = expected number of eventse = base of the natural logarithm system (2.71828.)!Xe)X(PxBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-47Poisson Distribution CharacteristicsMeanVariance and Standard Deviation 2 where = expected number of eventsBasic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 5-48Using Poisson TablesX 0.100.200.300.400.500.600.700.800.90012345670.90480.09050.0045