Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Probability / 9 / a / iv / Solution

Because $Z=e^{-Y}>0$, its cumulative distribution function is zero for $z\le 0$. For $z>0$, the transformation is decreasing, and $Y$ has a continuous cumulative distribution function, soThus the exponential transform of a standard Gumbel variable givesThis is the exponential distribution with rate $1$.