A compound Poisson random variable $S$ is defined as: $S=\displaystyle\sum^N_{i=1}X_i,$ where $N$ is a random draw from a Poisson distribution with intensity parameter $\lambda$, and $X_i$ are independent identically distributed random variables.
What is $\operatorname{Cov}(S,N)$?
$N$ is a Poisson random variable with mean $\lambda $. $S$ is compound Poisson random variable with Poisson parameter $\lambda $ and component distribution $F$.
$X_i$ where i =1,2,...is a sequence of iid r. v.s having distribution $F$