Standard uniform distribution (from http://www.math.wm.edu ...

Standard uniform distribution ( leemis/chart/ )The shorthandX U(0,1)is used to indicate that the random variableXhas the Standard uni-form distribution with minimum 0 and maximum 1. A Standard uniform random variableXhasprobability density functionf(x) =10<x< Standard uniform distribution is central to random variate generation. The probability densityfunction is illustrated (x)The cumulative distribution function on the support ofXisF(x) =P(X x) =x0<x< survivor function on the support ofXisS(x) =P(X x) =1 x0<x< hazard function on the support ofXish(x) =f(x)S(x)=11 x0<x< cumulative hazard function on the support ofXisH(x) = lnS(x) = ln(1 x)0<x< inverse distribution function ofXisF 1(u) =u0<u< median ofXis 1 moment generating function ofXisM(t) ={1t=0et 1tt6=0 The characteristic function ofXis (t) ={1t=0eit 1itt6=0 The population mean, variance, skewness and kurtosis ofXareE[X] =12V[X] =112E[(X )3]=0E[(X )4]= verification:The APPL statementsX := UniformRV(0,1);Mean(X);Variance(X);Skewn ess(X);Kurtosis(X);MGF(X).}}

The standard uniform distribution is central to random variate generation. The probability density function is illustrated below. 0 1 0 1 x f(x) The cumulative distribution function on the support of X is F(x)=P(X ≤x)=x 0 <x <1. The survivor function on the support of X is S(x)=P(X ≥x)=1−x 0 <x <1.

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Transcription of Standard uniform distribution (from http://www.math.wm.edu ...

1 Standard uniform distribution ( leemis/chart/ )The shorthandX U(0,1)is used to indicate that the random variableXhas the Standard uni-form distribution with minimum 0 and maximum 1. A Standard uniform random variableXhasprobability density functionf(x) =10<x< Standard uniform distribution is central to random variate generation. The probability densityfunction is illustrated (x)The cumulative distribution function on the support ofXisF(x) =P(X x) =x0<x< survivor function on the support ofXisS(x) =P(X x) =1 x0<x< hazard function on the support ofXish(x) =f(x)S(x)=11 x0<x< cumulative hazard function on the support ofXisH(x) = lnS(x) = ln(1 x)0<x< inverse distribution function ofXisF 1(u) =u0<u< median ofXis 1 moment generating function ofXisM(t) ={1t=0et 1tt6=0 The characteristic function ofXis (t) ={1t=0eit 1itt6=0 The population mean, variance, skewness and kurtosis ofXareE[X] =12V[X] =112E[(X )3]=0E[(X )4]= verification:The APPL statementsX := UniformRV(0,1);Mean(X);Variance(X);Skewn ess(X);Kurtosis(X);MGF(X).}}