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SCC GBS 221 - Chapter 05 Discrete Probability Distributions

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1Chapter 5Discrete Probability Distributions2Random Variablen A process whichn has more than one possible outcomen you don’t know in advance which outcome will occurn 2 major types (Ch 1)n continuousn can take on any value along relevant portion of measurement scalen can’t list every possible outcomen egs: height, weight, agen discreten can take on one of a finite number of values along the relevant portion of a measurement scalen can list every possible outcomen egs: # absent, # defective, # salesn Hoover assignment3Probability Distributionsn Describe the possible outcomes of a random variable & their associated probabilities of occurrence2 major typesn Continuous (Chs 6 & 8)n Normal, tn Discrete (Ch 5)n Uniform, Binomial, Poisson24f(x) > 0 Σf(x) = 1Discrete Probability Distributionsn Can take on one of a finite number of values along the relevant portion of a measurement scalen can list every possible outcomen Hoover examplen Probability Mass Function (nib)n the function that defines a discrete probability distributionn can be plottedn Required conditions for a discrete probability function:5n The simplest discrete probability distributionn Applies when the possible values of the random variable are all equally likely to occurn eg: a fair dien PMFDiscrete Uniform Probability Distributionwhere:n = the number of values the random variable may assumep(x) = 1/nx p(x)0 .16671 .16672 .16673 .16674 .16675 .16676 .16671.0000.05.10.15.20.251 2 3 4 5 6x = Die valueProbability6Summarizing Discrete Probability Distributionsn Skewn Expected Valuen weighted mean, long run average value of the random variablen Variancen dispersion among the possible values of the random variablep(x)][x = E(x) =µ ∑Computational formulan Hoover assignment37Bernoulli Trialn A process whichn has two mutually exclusive outcomes: success and failuren P(success) is constant over a series of trialsn Examplesn a coin tossn each Guess Your Best quiz questionn each Hoover sales call8Binomial Distributionn Used to determine the probability of a given # of successes in a given # of BTRIALSn Although the result of any individual trial is uncertain, the Binomial distribution accurately predicts the distribution of the number of successes over a series of BTRIALSn Discrete... why?9Binomial Formulan When involved with a series of BTRIALS, the corresponding probability tree has a special structurewhich allows streamlined calculationsp)-(1 px)!-(nx!n! = p)n,|(xPx)-(nXBn Hoover assignmentn Convenience Store | IRS | Basketball | Financial Aid410Binomial Tables Bookletn n=1, 2, 3, 4,.., 20 p=.01, .05, .10, .15,..., .95n Hoover scenario: n = 3, p = .20n PB(X=0 | n=3, p=.20)=n PB(X=1 | n=3, p=.20)=n PB(X=2 | n=3, p=.20)=n PB(X=3 | n=3, p=.20)=n Convenience Store | IRS | Basketballn Ch5 Wks 1, problem 3a-c11Mean And Variance Of Binomial Distributionn Because of Binomial distribution’s special structure:n Hoover assignmentn Convenience Store | IRS | Basketballnp = µ =(x)EBB12Binomial DistributionShape (Ch7)n The value of p determines the distribution’s skewnwhen p>.50 BINO is negatively skewedPB(x | n=5, p=.80)nwhen p<.50 BINO is positively skewedPB(x | n=5, p=.20)nwhen p=.50 BINO is symmetricalPB(x | n=5, p=.50)E(X) = 5(.2) =1σ2= 5(.2)(.8) = 0.8E(X) = 5(.5) =2.5σ2= 5(.5)(.5) = 1.25E(X) = 5(.8) = 4σ2= 5(.8)(.2) = 0.8n Ch5 Wks 1, problem 3a-c513Binomial DistributionShape (Ch7)n As the value of n increases, the BINO distribution becomes more symmetrical, regardless of pn as n ↑, the Binomial distribution approaches the Normal distribution PB(x | n=5, p=.35) PB(x | n=20, p=.35) PB(x | n=50, p=.35)n When np > 5 and n(1-p) > 5, the Binomial distribution can be approximated by a Normal distribution14Poisson Distributionn Used to determine the probability of a given number of successes which occur over a continuum of time or spacen A discrete distribution... why?n Two major assumptions:n probability of occurrence for an event is constant for any two intervals of equal lengthn the occurrence/nonoccurrence in any interval is independent of the occurrence/nonoccurrence in any other intervaln Must know (or estimate) µ , which represents _____n Caution when question’s units of measurement don’t match the mean’s units of measurement … rescale the mean to match the questionx!eµ = µ)|(xP-µxp15Poisson Distributionn Poisson probability Tables n Maytag Problemn Because of Poisson’s special structure (nib)n Maytag:E(calls/hr) = 1.5 σ2(calls/hr) = 1.5µ


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