Variance of $X_i / \sum\limits_{j=1}^n X_j$

Question

Suppose that $X_1 ,X_2,...X_N$ are independent identically distributed positive random variables. How to find the variance of $\frac{X_i}{\sum\limits_{j=1}^n X_j}$, $i \in 1, 2, \ldots, n$?

Answer 1

I don't think this question can be fully answered without more information. But we can find a few quantities. Let $n=N$ and define $Y_i={{X_i} \over {\sum_{j=1}^N X_j}}.$ Let the variance of $Y_i$ be $\sigma^2$ and the covariance be $\sigma_{Y_iY_j}=\sigma_{12}.$ We can do this since the $Y_i$ are identically distributed. Calling their sum $Z,$ we know $Z=1$ so $E[Y_i]= {{1 \over N}}.$ The variance of $Z$ is zero, so we also know $$\sigma^2_Z=\sum^N \sigma^2 + 2 \sum_{i < j} \sigma_{Y_iY_j}=0.$$ This yields $$N \sigma^2 + N(N-1)\sigma_{12}=0$$ so we have $$\sigma_{12}={{-\sigma^2} \over {N-1}}$$ The correlation $\rho$ between $Y_i$ and $Y_j$ is then $$\rho ={{\sigma_{12}} \over {\sigma^2}}={{-1} \over {N-1}}.$$ This is far as I've been able to get. I've not been able to find any expression for $\sigma^2$ - perhaps another can provide further insight.