= Exponential transform of a standard Gumbel variable
{title2=$Y\sim\mathrm{Gumbel}(0,1)\implies e^{-Y}\sim\operatorname{Exp}(1)$}
If $Y$ has <cumulative distribution function> $F(y)=\exp(-e^{-y})$, put $Z=e^{-Y}$. For $z>0$, the decreasing transformation gives $\mathbb P(Z\le z)=1-F(-\ln z)=1-e^{-z}$, while this <probability> is zero for $z\le0$. Hence $Z$ has the rate-one <exponential distribution>. Conversely, if $Z$ has that <exponential distribution>, then $-\ln Z$ has the standard <Gumbel distribution>.
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