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Given: $Z=\sqrt{X^2+Y^2}, X\sim N(\mu_x,\sigma_x^2), Y\sim N(\mu_y,\sigma_y^2)$

What is the expected value of $Z$?

I'm specifically looking for the case where the $\mu_i$ are non-zero and $\sigma_i$ are different, so it's neither a generalized Chi nor a non-central Chi, as far as I could tell.

A more general form for the norm of N normal variables would also be nice.

Implication
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  • Hint: Set up the standard integral, and convert it into the polar form $(x,y) \mapsto (r\cos \theta, r \sin \theta)$ – stochasticboy321 Nov 05 '15 at 20:06
  • Believe me, I've tried the obvious.

    Transforming into polar coordinates won't really help since $X,Y$ aren't centered, and the argument inside the exponent ends up as $-(\frac{(\mu_x-r\cos(\theta))^2}{2 \sigma_x}+\frac{(\mu_x-r\sin(\theta))^2}{2 \sigma_y})$. Mathematica is having trouble solving even explicit cases with unequal $\mu_i,\sigma_i$, let alone finding a general solution.

     – Implication Nov 06 '15 at 12:17
  • Ach, yes, that was daft of me, I didn't read the question properly. – stochasticboy321 Nov 07 '15 at 00:37
  • I found a partial answer here: http://math.stackexchange.com/questions/605267/length-of-gaussian-random-vector – stochasticboy321 Nov 07 '15 at 01:16

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