Chapter 3.4, Problem 12E

Let X have an exponential distribution with $\theta =1$; that is, the pdf of X is $f\left(x\right)=e^{-x},0<x<\infty$.

Let T be defined by $T=\ln X$, so that the cdf of T is $G\left(t\right)=P\left(\ln X\le t\right)=P\left(X\le e^t\right)$.

(a) Show that the pdf of T is $g\left(t\right)=e^t{e}^{-e^t},-\infty <t<\infty ,$ which is the pdf of an extreme-value distribution.

(b) Let W be defined by $T=\alpha +\beta \ln W$, where $-\infty <\alpha <\infty$ and $\beta >0$.

Show that W has a Weibull distribution.