Chapter 5.4Binomial DistributionA special application of a discrete probability distribution is thebinomial distribution.To start, introduce the random variablethat takes two outcomes:BXx = 1“success”x = 0“failure”The probability distribution function ofis:BXfor 0 <p< 1(the probability of success)p)X(PB==1p1)X(PB−==0This is known as the Bernoulli distribution.The mean and variance are calculated as:ppp)x(Px)X(ExBXB=−⋅+⋅===μ∑)(101)p1(ppppppp)x(Px)X(E)X(Var222x22X2BBB−=−=−−⋅+⋅=−=μ−=∑)(1(0)(0)(1)(1)Chapter 5 17
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Now consider that a random experiment with the outcome of successor failure is repeatedntimes.Each trial produces success or failure with probabilitiespand (1−p)respectively. Assume independence so that the result of one trialdoes not influence the result of any other trial.Let the random variableXbe the number of successes inntrials.The probability distribution function ofXis defined as:)(P)x(Ptrialstindependenninsuccessesx=forx= 0, 1, 2, . . . ,nThis is known as thebinomial distribution.Chapter 5 18