What is the expected value of the division of a random variable by a sum of random variables? With $X_1$, $X_2$ and $X_3$ being independent random variables, how can I compute $\mathbb{E}\left[ \frac{X_1}{X_1+X_2+X_3}\right]$?
Is $\mathbb{E}\left[ \frac{X_1}{X_1+X_2+X_3}\right] = \frac{\mathbb{E}\left[X_1\right]}{\mathbb{E}\left[X_1\right]+\mathbb{E}\left[X_2\right]+\mathbb{E}\left[X_3\right]}$? If not, how is it  calculated?
Thank you in advance for any clarification.
 A: There is a famous "folk theorem" residual to my undergrad classes, namely that
$$\mathbb E[X/Y] = \mathbb E[X] \big/ \mathbb E[Y]$$
and it is often permeates during exams, as it makes computing much easier. Sadly, the equality does not hold in general, even when $X$ and $Y$ are independent (due to Jensen's inequality).
In the case of $\mathbb E[X_1/X_1+X_2+X_3]$, numerator and denominator are dependent, which usually makes the computation more difficult. However, in the very special case when the three $X_i$'s are iid, $X_i/X_1+X_2+X_3$ has the same distribution for all three $i$'s and this leads to an obvious conclusion concerning the expectation of any of them. Assuming this expectation exists, of course. A counterexample is provided by a triplet of Normal variables (see Marsaglia's paper in connection).
As a special case where the identity works, take the Dirichlet $\mathcal D(\alpha_1,\ldots,\alpha_d)$ distribution, whose expectation is
$$\mathbb E[Y_i]=\alpha_i\Big/\sum_{j=1}^d \alpha_j$$
One representation of a Dirichlet random vector $(Y_1,\ldots,Y_d)$ is
$$Y_i=\frac{X_i}{X_1+\ldots+X_d}\qquad X_i\sim\mathcal G(\alpha_i,1)$$
where the $X_i$'s are independent. In that case,
$$\mathbb E[Y_i]=\mathbb E[X_i\big/X_1+\ldots+X_d]=
\mathbb E[X_i]\big/\mathbb E[X_1+\ldots+X_d]$$
