TITLE: Expectation with square root
QUESTION [5 upvotes]: I don't know how to calculate the expectation when there is some square root in the expression. My problem is this: we have three real random variables $X,Y,Z$, independent and with standard normal distribution $N(0,1)$ and we want to calculate
$$E\left(\frac{1}{2} \left(X + Z + \sqrt{X^2 + 4 Y^2 - 2 XZ + Z^2}\right)\right).$$
How can this be done?
Thanks.

REPLY [4 votes]: @jens
The line should read as
$\begin{align}
E
&= \frac{1}{2}\mathsf{E}\left[\sqrt{4Y^2 + (\sqrt{2}X)^2}\right] \\ \\
&= \frac{\sqrt{2}}{2}\mathsf{E}\left[\sqrt{2Y^2 + X^2}\right]
\end{align}$
(I see you have done so.)
Now Mathematica can do the calculation easily and get:
$E=\frac{EllipticE(-1)}{\sqrt{\pi }}\approx 1.0776657899830802$
which agrees with simulations.