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Suppose you have three independent random variables $x$, $y$, and $z$ that are distributed as follows:

$x$ ~ $N(0,\sigma_x)$, $y$ ~ $N(0,\sigma_y)$, and $z$ ~ $N(z_{average},\sigma_z)$.

I am trying to find the expected value and variance of the quantity $P=\sqrt{x^2 + y^2 +z^2}$. Since $\sigma_P^2 = E(P^2)-E(P)^2$, and $E(P^2)=\sigma_x^2+\sigma_y^2+\sigma_z^2+z_{average}^2$, I only need the expected value of this quantity. I have looked at some related questions and see that it should be of the form $$\iiint \sqrt{x^2+y^2+z^2}\frac{1}{\sigma_x\sqrt{2\pi}}e^\frac{-x^2}{2\sigma_x^2}\frac{1}{\sigma_y\sqrt{2\pi}}e^\frac{-y^2}{2\sigma_y^2}\frac{1}{\sigma_z\sqrt{2\pi}}e^\frac{-(z-z_{average})^2}{2\sigma_z^2}\,dx\,dy\,dz$$ But I have no idea how to evaluate this integral. I understand that with two dimensions elliptical integration can be used, but what kind of method is there to evaluate this triple integral?

Thanks in advance.

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  • $\begingroup$ What's $z_{\mathrm{average}}$? $\endgroup$ – user217285 Aug 24 '17 at 22:48
  • $\begingroup$ $z_{average}$ represents the mean of random variable z. Its some non-zero number. $\endgroup$ – paulinho Aug 25 '17 at 1:01
  • $\begingroup$ Check out this and this. $\endgroup$ – user3658307 Aug 25 '17 at 15:42
  • $\begingroup$ But how would the answer be changed if the means were non-zero or if the variances differed? $\endgroup$ – paulinho Aug 25 '17 at 20:41
  • $\begingroup$ Yes, it's tough. Btw you need to use "@" or I wont see your comment. Here are some more links: here, here, here, and here. $\endgroup$ – user3658307 Aug 25 '17 at 21:02

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