Expectation of random variables ratio

Solution 1:

Hint: In addition to the linearity property you mentioned, use the following facts:

$1$) By symmetry we have $E\left(\frac{X_i}{\sum}\right)=E\left(\frac{X_j}{\sum}\right)$.

$2$) $E\left(\frac{\sum}{\sum}\right)=E(1)=1$. This, $1$), and linearity forces $E\left(\frac{X_i}{\sum}\right)=\frac{1}{n}$.

Existence is not a problem since $0\lt \frac{X_i}{\sum}\lt 1$.

Solution 2:

Page 2 gives the solution to this:

http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-436j-fundamentals-of-probability-fall-2008/recitations/MIT6_436JF08_rec05.pdf

The crux of the problem is covered in the answer by André Nicolas.