Answer
$\approx 0.982$
Work Step by Step
We are given that
$p(x)=Ce^{-x} e^{-e^{-x}}$
Suppose that
$a =-e^{-x} \implies da=e^{-x} dx$
Consider
$I=\int_{-\infty}^{\infty} Ce^{-x} e^{-e^{-x}} dx \\=C \int_{-\infty}^{\infty} e^{u} du\\=C \lim\limits_{R \to \infty}(e^{-e^{-R}}-e^{-e^{R}}) \\=C(1-0)\\=C$
We will find
$P(-4 \leq X \leq 4)=\int_{-4}^{4} p(x) \ dx\\=\int_{-4}^4 e^{-x} e^{-e^{-x}} \\=[e^{-e^{-x}}]_{-4}^4 \\=e^{-e^{-4}}-e^{-e^{4}} \approx 0.982$
