Let
$$Y_1\sim \operatorname{Poisson}(\lambda_1)\\Y_2\sim \operatorname{Poisson}(\lambda_2),$$ where $Y_1$ and $Y_2$ are independent, and $\lambda_1, \lambda_2>0$.
What is the variance of $$\frac{Y_1} {Y_1+Y_2}?$$
Or more general, what is the variance of $$\frac{X_1} {X_1+ \cdots +X_n},$$where $X_i\sim \operatorname{Poisson}(\lambda_i)$, $i=1,...,n$?
EDIT:
This idea came to mind: It is well-known that if $Y_1\sim \operatorname{Poisson}(\lambda_1)$ and $Y_2\sim \operatorname{Poisson}(\lambda_2)$, where $Y_1$ and $Y_2$ are independent, then $Y_1|Y_1+Y_2\sim \operatorname{Binomial}(y_1+y_2,p)$. Since the variance of a Binomially distributed random variable is $np(1-p)$, I just need to find the MLE of $p$.
Could this be a solution to my problem?