Gaussian-over-Rayleigh inclination distribution ($F_{\alpha }(A)=\frac{\Gamma A}{\sqrt{{\Gamma }^2{A}^2+{\Lambda }^2}}$)

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 $A\ge 0$ and density ${\Lambda }^2\Gamma ({\Gamma }^2{A}^2+{\Lambda }^2{)}^{-3/2}$. Indeed, conditioning on $D$ gives $\mathbb{P}(c\ge |D|/A)=\exp [-D^2/(2{\Gamma }^2{A}^2)]$, 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 $\Lambda /\Gamma$.