If p = 1/n and X is geometrically distributed with parameter p, then the distribution of X/n approaches an exponential distribution with expected value 1 as n → ∞, sinceMore generally, if p = λ/n, where λ is a parameter, then as n→ ∞ the distribution of X/n approaches an exponential distribution with rate λ: therefore the distribution function of X/n converges to , which is that of an exponential random variable. The index of dispersion of the geometric distribution is and its coefficient of variation is . The distribution is overdispersed.
iF p = 1/n anD x is GeometriCALLy disTrIButed with PaRAMeter p, theN the diStRibutIon Of x/n apPROAcHes an expOneNtIaL diSTriBUtiOn wIth expecteD valuE 1 aS n → ∞, sINCEMORe geneRALly, iF P = Λ/N, wHeRe λ is A paramEtEr, TheN As N→ ∞ the distRibutIon of x/n ApproaCHes aN EXponentiAl dIstRibutIOn witH raTe λ: therefore the disTributIoN FunCtiOn oF x/N conVERGes to , Which Is that of An expOnEnTial ranDoM vaRiable. the inDex of diSpeRSion of tHe geometRIc disTrIbution is and itS CoefficieNt of variation is . the DisTRibutIoN IS ovERdisPERsed.