3D spherical coordinates change

Question

Let $X \sim N(0,a^2)$, $Y \sim N(0,b^2)$, $Z \sim N(0,c^2)$ be independent normal distributions.

Let $x= a\rho \sin\phi \cos\theta$, $y= b\rho \sin\phi \sin\theta$, $z= c\rho \cos\phi$. Using the above change of variables, calculate

$$\Bbb E\left[\left(\frac{X^2}{a^2}+\frac{Y^2}{b^2}+\frac{Z^2}{z^2}\right)^{-1/2}\right]$$

I understand everything in terms of evaluating the triple integral, but I do not understand how to obtain the limits of integration wrt. $\rho$, $\phi$, $\theta$. What's the shape you're transforming going to be?

Answer 1

This probably means to "parametrize" 3-space with spherical polar coordinates. (The quotes are because it's a bad parameterization at the origin, just as lat-lon coordinates on the unit sphere are bad at the N/S pole and along the international dateline).

If that's the case, then \begin{align} 0 & \le \rho < \infty \\ 0 & \le \theta < 2\pi \\ 0 &\le \phi \le \pi \end{align} would be a typical domain. Some folks might choose $-\pi \le \theta \le \pi$, I suppose. Same result either way, because of periodicity of sine and cosine.

Answer 2

I suggest you a preliminary before proceeding. Define $$ p=X/a\ \ \ ,\ \ \ q=Y/b\ \ \ ,\ \ \ r=Z/c, $$ where $p,q,r\sim\mathcal{N}(0,1)$. So you seek to find $$ \mathbb{E}\{(p^2+q^2+r^2)^{-\frac{1}{2}}\}. $$ Now the bounds must span whole the space since $(X,Y,Z)$ do.