Given two positive constants $c\_1,c\_2$ and two independent standard normal random variables $a,b$, how to calculate the following expected value
$$
\mathbb{E}\left[\frac{a^2}{c\_1a^2+c\_2b^2}\right]
$$
Thank you.



Expected value of a ratio of squared normal and linear combination of squared normal
$\newcommand\E{\mathscr E}$The expectation in question is
$I/c\_1$, where
$$I:=E\frac{a^2}{a^2+cb^2},\quad c:=c\_2/c\_1>0.$$
In turn, using polar coordinates, we get
$$I=\frac1{2\pi}\int\_0^{2\pi}\frac{\cos^2 t\, dt}{\cos^2 t+c\sin^2 t},\quad c:=c\_2/c\_1>0.$$
Further, writing
$$I=\frac4{2\pi}J$$
for
$$J:=\int\_0^{\pi/2}\frac{\cos^2 t\, dt}{\cos^2 t+c\sin^2 t}$$
and using the standard substitution $t=\arctan u$, so that $\cos^2 t=1/(1+u^2)$ and $\sin^2 t=u^2/(1+u^2)$, we have
$$J=\int\_0^\infty\frac{du}{(1+cu^2)(1+u^2)}.$$
Using partial fraction decomposition to compute $J$ and collecting the pieces, we finally get that the expectation in question is
$$\E(c\_1,c\_2):=\frac1{c\_1+\sqrt{c\_1c\_2}}.$$




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"Sanity" checks: (i) $\E(tc\_1,tc\_2)=\E(c\_1,c\_2)/t$ for all real $t>0$; (ii) $\E(1,0)=1$; (iii) $\E(1,1)=1/2$; (iv) $\E(c\_1,c\_2)\to\infty$ as $c\_1\downarrow0$.


