We have $X_1, X_2, \ldots$ – independent random variables with uniform distribution on interval $[0,1]$. $N$ – random variable independent of $X_1,X_2,\ldots$ with Poisson distribution with $\lambda=2$. I have to calculate $\operatorname{cov}({V_N, N})$ where $V_N=\frac{X_1}{S_N}$ where $S_N=X_1+\cdots+X_{N+1}$.
$$\operatorname{cov}(V_N,N)=E(V_NN)-E(V_N)E(N)$$
and now I have problem with calculation of $E(V_NN)$ and $E(V_N)$
I write $$E(V_N)=E(E(V_N\mid N=n))$$
but then I dont know how to calculate $E(V_N\mid N=n)=E\left(\frac{X_1}{X_1+\cdots+X_{n+1}}\right)$
Best Answer
There are surely typos in your post as you have the same notation for both uniform rvs and Poisson rvs. Either way, the last expectation is not that hard.
You see, $X_1,...,X_n$ are iids, hence . $ \mathbb{E}\left( \frac{X_1}{X_1+...+X_n} \right) = \mathbb{E}\left( \frac{X_2}{X_1+...+X_n} \right)=etc$
As their sum is clearly equation to 1, the desired expectation is equal to $\frac{1}{n+1}$