2/26/20191Department of Industrial & Systems Engineering, HCMUTEmail: [email protected]Nguyen VP Nguyen, Ph.D.February 26, 2019Discrete probability distributionsBinomial distributionMultinomial distributionPoisson distributionHypergeometric distributionContinuous probability distributionsNormal distributionStandard normal distributionGamma distributionExponential distributionChi square distributionLognormal distributionWeibull distributionTAXONOMY OF PROBABILITY DISTRIBUTIONS
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2/26/20192USAGE OF PROBABILITY DISTRIBUTIONDistribution(discrete/continuous)function is widely used in simulation studies.•A simulation study uses a computer to simulate a real phenomenon or process asclosely as possible.•The use of simulation studies can often eliminate the need of costly experimentsand is also often used to study problems where actual experimentation isimpossible.Examples:1)A study involving testing the effectiveness of a new drug, the number of curedpatients among all the patients who use such a drug approximately follows abinomial distribution.2)Operation of ticketing system in a busy public establishment (e.g., airport), thearrival of passengers can be simulated usingPoisson distribution.BINOMIAL DISTRIBUTION•In many situations, an outcome has only two outcomes:successandfailure.Such outcome is called dichotomous outcome.•An experiment when consists of repeated trials, each with dichotomousoutcome is calledBernoulli process. Each trial in it is called aBernoulli trial.Example 4.5:Firing bullets to hit a target.•In a Bernoulli process, we define a random variableX≡ thenumber of successes intrials.This a random variable obeys the binomial probability distribution, if theexperiment satisfies the following conditions:1)The experiment consists ofntrials.2)Each trial results in one of two mutually exclusive outcomes, one labelled a“success”and the other a“failure”.3)The probability of a success on a single trial is equal to. The value ofremainsconstant throughout the experiment.4)The trials are independent.
2/26/20193BINOMIAL DISTRIBUTIONThefunctionforcomputingtheprobabilityforthebinomialprobabilitydistribution is given by݂ݔൌ݊!ݔ!݊ െ ݔ !௫ሺ1 െ ሻି௫for x = 0, 1, 2, …., nHere,݂ݔൌ ܲܺ ൌ ݔ ,whereܺdenotes “the number of success” andܺ ൌ ݔdenotes the number of success inݔtrials.Definition: Binomial distributionCRITERIA FOR USING BINOMIAL DISTRIBUTIONSThe binomial distribution is used to model the probabilities of occurrences when specific rules are met.•Rule #1: There are only two mutually exclusive outcomes for a discrete random variable (i.e., success or failure).