= Laplace ratio-of-uniforms envelope
{c}
{title2=$a=1,\quad b_1=-4e^{-5/4},\quad b_2=4e^{-3/4}$}
For the unnormalized <Laplace distribution> <probability density function> $h(x)=e^{-|x-1|/2}$, the <ratio-of-uniforms method> uses $0<u\leq\sqrt{h(v/u)}$. The maximum of $h$ is one. On $x\leq0$, the maximum of $x^2h(x)$ occurs at $x=-4$ and equals $16e^{-5/2}$. On $x\geq0$, it occurs at $x=4$ and equals $16e^{-3/2}$. Thus the displayed rectangle contains the region. Its area is $4(e^{-3/4}+e^{-5/4})$, while the target region has area $\tfrac12\int h=2$. Uniform proposals in the rectangle, accepted when $u^2\leq h(v/u)$, therefore return the correct <Laplace distribution> with acceptance <probability> $[2(e^{-3/4}+e^{-5/4})]^{-1}$.
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