Solution
= Solution
The proposed <cumulative distribution function> satisfies $0<F(x)<1$ and is smooth. Its <derivative> is
$$
F'(x)=e^{-x}\exp(-e^{-x})>0,
$$
so it is increasing and continuous, hence right-continuous. As $x\to-\infty$, $e^{-x}\to\infty$ and $F(x)\to0$; as $x\to\infty$, $e^{-x}\to0$ and $F(x)\to1$. Thus \b[it has all the required <cumulative distribution function> properties], and defines the standard <Gumbel distribution>.