Bernoulli distribution X - William & Mary

Bernoulli distribution ( leemis/chart/ )The shorthandX Bernoulli (p)is used to indicate that the random variableXhas the Bernoullidistribution with parameterp, where 0<p<1. A Bernoulli random variableXwith successprobabilityphas probability mass functionf(x) =px(1 p)1 xx=0,1for 0<p<1. The Bernoulli distribution is associated with the notion of aBernoulli trial, whichis an experiment with two outcomes, generically referred toassuccess(x=1) andfailure(x=0).The cumulative distribution function ofX Bernoulli (p)isF(x) =P(X x) = 0x<01 p0 x<11x survivor function ofXisS(x) =P(X x) = 1x 0p0<x 10x> hazard function ofXon the support ish(x) =f(x)S(x)= 1 px=01x= cumulative hazard function ofXonx 1 isH(x) = lnS(x) = 0x 0 lnp0<x inverse distribution function ofXisF 1(u) = 00<u<1 p11 p u< median ofXis 0 if 0<p 1/2 and 1 if 1/2<p<1.

The Bernoulli distribution is associated with the notion of a Bernoulli trial, which is an experiment with two outcomes, generically referred to as success (x =1) and failure (x =0). The cumulative distribution function of X ∼Bernoulli(p)is

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Transcription of Bernoulli distribution X - William & Mary

1 Bernoulli distribution ( leemis/chart/ )The shorthandX Bernoulli (p)is used to indicate that the random variableXhas the Bernoullidistribution with parameterp, where 0<p<1. A Bernoulli random variableXwith successprobabilityphas probability mass functionf(x) =px(1 p)1 xx=0,1for 0<p<1. The Bernoulli distribution is associated with the notion of aBernoulli trial, whichis an experiment with two outcomes, generically referred toassuccess(x=1) andfailure(x=0).The cumulative distribution function ofX Bernoulli (p)isF(x) =P(X x) = 0x<01 p0 x<11x survivor function ofXisS(x) =P(X x) = 1x 0p0<x 10x> hazard function ofXon the support ish(x) =f(x)S(x)= 1 px=01x= cumulative hazard function ofXonx 1 isH(x) = lnS(x) = 0x 0 lnp0<x inverse distribution function ofXisF 1(u) = 00<u<1 p11 p u< median ofXis 0 if 0<p 1/2 and 1 if 1/2<p<1.

2 The mode ofX, denoted bym, ism= 00<p<1/211/2<p< moment generating function ofXisM(t) =E etX = (1 p)+pet <t< .The characteristic function ofXis (t) =E eitX = (1 p)+peit <t< .The population mean, variance, skewness, and kurtosis ofXareE[X] =pV[X] =p(1 p)E" X 3#=1 2ppp(1 p)E" X 4#=3p2 3p+1p(1 p).1 APPL verification:The APPL statementsX := BernoulliRV(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

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