Transcription of Geometric distribution (from X - William & Mary
1 Geometric distribution ( leemis/chart/ )The shorthandX Geometric (p)is used to indicate that the random variableXhas the geometricdistribution with real parameterpsatisfying 0<p<1. A Geometric random variableXwithparameterphas probability mass functionf(x) =p(1 p)xx=0,1,2,..The Geometric distribution can be used to model the number offailures before the first success inrepeated mutually independent Bernoulli trials, each with probability of successp. For example,the Geometric distribution withp=1/36 would be an appropriate model for the number of rolls ofa pair of fair dice prior to rolling the first double six.
2 The Geometric distribution is the only discretedistribution with the memoryless property. The only continuous distribution with the memorylessproperty is the exponential distribution . The probabilitymass function withp=1/36 is (x)The cumulative distribution function on the support ofXisF(x) =P(X x) =1 (1 p)x+1x=0,1,2,..The survivor function isS(x) =P(X x) = (1 p)xx=0,1,2,..The hazard function ish(x) =f(x)S(x)=px=0,1,2,..The cumulative hazard function isH(x) = lnS(x) = ln((1 p)x)x=0,1,2,..1 The inverse distribution function ofXisF 1(u) = ln(1 u)ln(1 p) 0<u< moment generating function ofXisM(t) =E[etX]=p1 (1 p)ett< ln(1 p).
3 The population mean, variance, skewness, and kurtosis ofXareE[X] =1 ppV[X] =1 pp2E[(X )3]=2 p 1 pE[(X )4]=9+p21 second parameterization of the Geometric distribution exists with the support starting at 1. Forthis parameterization the probability mass function isf(x) =p(1 p)x 1x=1,2,..This is the probability mass function used in verification:The APPL statementsX := GeometricRV(p);CDF(X);SF(X);HF(X);CHF(X) ;IDF(X);Mean(X);Variance(X);Skewness(X); Kurtosis(X);MGF(X);verify the cumulative distribution function, survivor function, hazard function, cumulative hazardfunction, inverse distribution function, population mean, variance, skewness, kurtosis, and momentgenerating
