Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}\_{M},\sigma\_{x}^{2}\textbf{I}\_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}\_{M},\sigma\_{y}^{2}\textbf{I}\_{M})$ are independent, what would be the expectation


$$\mathbb{E} \left[ \left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2 \right], $$
where $\mathcal{CN}(.,.)$ is the complex normal random variable.



Expectation of $\left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2$, with complex Gaussians?
Using the same trick from [another answer](https://mathoverflow.net/a/322939/114668), as well as the trace trick and $E[1/\Vert z\Vert^2]$ from [yet another answer](https://mathoverflow.net/a/323414/114668), we find


$$
\frac{\sigma\_x^2 \, \sigma\_y^2}{(\sigma\_x^2+\sigma\_y^2)^2} \, \frac{1}{M-1} + \frac{(\sigma\_x^2)^2}{(\sigma\_x^2+\sigma\_y^2)^2} \;.
$$


