Distribution of the square root of the sum of two i.i.d. random variables.

Question

For the random variable $Z=\sqrt{U}=\sqrt{X^2+Y^2}$ where $X$ and $Y$ are two i.i.d. random variables with pdf

$$f_X(x)=\frac{1}{\pi \sqrt{1-x^2}}\,,\quad x\in[-1,1]$$

The probability density functions of $X$ and $X^2$ are easy to calculate, but the pdf of $U$ is difficult to calculate since the integral is hard.
I would like to get the expectation and variance of $Z$. Is there any way to estimate the expectation and variance as accurate as possible? Like Taylor expansion or something else?

Answer 1

First, $$\mathbb E\left[Z^2\right] = 2\mathbb E\left[X^2\right] = \frac2\pi\int_{-1}^{1} \frac{x^2}{\sqrt{1-x^2}}\mathrm d x = \frac2\pi\int_{0}^1 u^{\frac12}(1-u)^{-\frac12}\mathrm d u = \frac2\pi B\left(\frac32, \frac12\right) = 1.$$

Second,

\begin{align} \mathbb E\left[Z\right] &= \frac{4}{\pi^2}\int_{0}^1\int_{0}^1 \frac{\sqrt{x^2 + y^2}}{\sqrt{1 - x^2}\sqrt{1 - y^2}}\mathrm d x\mathrm d y\\ &\approx \frac4{n\pi^2}\sum_{k=0}^{n-1}\sum_{\ell = 0}^{n-1} \frac{\sqrt{k^2 + \ell^2}}{\sqrt{n^2 - k^2}\sqrt{n^2 - \ell^2}} \end{align}