I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius..
$X \sim \mathcal N(\mu_1,\sigma_1)$,
$Y \sim \mathcal N(\mu_2,\sigma_2)$,
$Z \sim \mathcal N(\mu_3,\sigma_3)$
They are independent! Seems easy so far, right?
You tell me, is this easy?
Compute $$\mathbb E\left[\frac{X}{X+Y+Z}\right]$$ and $$\operatorname{Var}\left[\frac{X}{X+Y+Z}\right].$$
Edit:
As someone in the comments pointed out this quantity is not defined. What about if I say:
$0<X<1$,
$0<Y<1$,
$0<Z<1$
So that the p.d.f.s of X,Y, and Z are divided by
$\Phi(1-\mu_1/\sigma_1)-\Phi(-\mu_1/\sigma_1),\Phi(1-\mu_2/\sigma_2)-\Phi(-\mu_2/\sigma_2),\Phi(1-\mu_3/\sigma_3)-\Phi(-\mu_3/\sigma_3)$, respectively (to make the sums under the curves 1 - did I do that right?).
Sorry if this is a ridiculous question.