Let X have the pdf fx 1 0 x 1 Show that the random varia

. Let X have the pdf f(x) = 1, 0 < x < 1. Show that the random variable Y = log(X) has an exponential distribution. What is the mean of Y ?

Solution

Given,f(x)=1,0<x<1

Y=-log(X)

x=e^-y

0<x<1

=>log(x)<0

=>y=-log(x)>0

dX/dY=-e^-y

J=|dX/dY|=e^-y

f(y)=1.e^-y

=>f(y)=e^-y ,y>0

Thus,Y~exp(1)

Therefore,mean(Y)=1