Evaluate the $Cov(\sum_{k=1}^{N}Y_k, N)$

Question

Let $Y_1, Y_2$... be i.i.d. random variables, and let $N$ be a nonnegative, integer valued random variable that is independent of $Y_1, Y_2$... Compute $$Cov(U, N)$$ where $U=\sum_{k=1}^{N}Y_k$

This is how far i came:

$$Cov(U, N)= E(UN) - E(Y)(E(N))^2$$

My problem is that i have no idea how to evaluate $E(UN)=E(N\cdot \sum_{k=1}^{N}Y_k$).

Any ideas?

Answer 1

$$\mathsf {Cov}(U,N) ~=~ \mathsf E(UN)-\mathsf E(U)\cdotp\mathsf E(N)$$

Use the Law of Iterated Expectation, Linearity of Expectation, and independence of $(Y_i)$ and $N$, and the identical independent distribution of $(Y_i)$.

For example: $\mathsf E(U) ~{= \mathsf E\Big(\mathsf E(\sum_{k=1}^NY_k\mid N)\Big)\\= \mathsf E\Big(\sum_{k=1}^N\mathsf E(Y_k\mid N)\Big)\\ = \mathsf E(N\,\mathsf E(Y_1))\\= \mathsf E(N)\cdot\mathsf E(Y_1)}$

Then likewise: $\mathsf E(UN) ~{= \mathsf E\Big(\mathsf E(N\,\sum_{k=1}^NY_k\mid N)\Big)\\ \ddots}$