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

$$\mathsf {Cov}(U,N) ~=~ \mathsf E(UN)-\mathsf E(U)\cdotp\mathsf E(N)$$
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: $\mathsf E(UN) ~{= \mathsf E\Big(\mathsf E(N\,\sum_{k=1}^NY_k\mid N)\Big)\\ \ddots}$
