Expected value of the distance to the center of a cube assume $X_1$, $X_2$ and $X_3$ are three independent variables with uniform distribution on (-1,1)
what is the expected value of $\sqrt{X_1^2+X_2^2+X_3^2}$?
My thought is that since they are independent, it is equivalent to calculate the expected value of $\sqrt{3X_1^2}$, which really means to calculate the expected value of $\sqrt{3|X_1|}$

Am i correct?　any hint is welcome. Thank you.

 A: Why not use a computer algebra system to do the nitty gritties for you? In this instance, the joint pdf of $(X,Y,Z)$ is $f(x,y,z)$:

and the expectation you seek is:

where Expect is a function from the mathStatica add-on to Mathematica (I should add I am one of the authors of the former ... but equally I am sure other packages could also do this). The answer is approximately 0.960592. 
By contrast, $E[\sqrt{3 X^2}] = \frac{\sqrt{3}}{2}  \approx 0.866$.
A: $\newcommand{\+}{^{\dagger}}%
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$\large\tt Hint:$
\begin{align}
\int_{-1}^{1}{1 \over 2}\int_{-1}^{1}{1 \over 2}\int_{-1}^{1}{1 \over 2}\,
\root{x^{2} + y^{2} + z^{2}}\,{\rm d}x\,{\rm d}y\,{\rm d}z
=
\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}
\root{x^{2} + y^{2} + z^{2}}\,{\rm d}x\,{\rm d}y\,{\rm d}z
\end{align}
