= Gaussian-over-Rayleigh inclination distribution
{c}
{title2=$F_\alpha(A)=\frac{\Gamma A}{\sqrt{\Gamma^2A^2+\Lambda^2}}$}
Let $c=\sqrt{a^2+b^2}$ with independent $a,b\sim N(0,\Gamma^2)$, and let $D\sim N(0,\Lambda^2)$ be independent of them. Then $c$ has <Rayleigh distribution> and $\alpha=|D|/c$ has the displayed <cumulative distribution function> for $A\ge0$ and density $\Lambda^2\Gamma(\Gamma^2A^2+\Lambda^2)^{-3/2}$. Indeed, conditioning on $D$ gives $\mathbb P(c\ge|D|/A)=\exp[-D^2/(2\Gamma^2A^2)]$, and a <Gaussian integral> completes the calculation. Independence is essential: choosing $D=(\Lambda/\Gamma)a$ has the same Gaussian marginal but makes the ratio bounded by $\Lambda/\Gamma$.
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