Proof A geometric random variable has the memoryless property if for all nonnegative integers and or equivalently The probability mass function for a geometric random variab le is 1 0 The probability that is greater than or equal to is 1 ID: 25687
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TheoremThegeometricdistributionhasthememoryless(forgetfulness)property.ProofAgeometricrandomvariableXhasthememorylesspropertyifforallnonnegativeintegerssandt,P(Xs+tjXt)=P(Xs)or,equivalentlyP(Xs+t)=P(Xs)P(Xt):TheprobabilitymassfunctionforageometricrandomvariableXisf(x)=p(1 p)xx=0;1;2;::::TheprobabilitythatXisgreaterthanorequaltoxisP(Xx)=(1 p)xx=0;1;2;::::SotheconditionalprobabilityofinterestisP(Xs+tjXt)=P(Xs+t;Xt) P(Xt)=P(Xs+t) P(Xt)=(1 p)s+t (1 p)t=(1 p)s=P(Xs);whichprovesthememorylessproperty.APPLveri cation:TheAPPLstatementssimplify((1-op(CDF(GeometricRV(p)))(s)[1])*(1-op(CDF(GeometricRV(p))(t)[1])));1-simplify(op(CDF(GeometricRV(p))(s+t)[1]));bothyieldtheexpression(1 p)s+t:1