Find the expectation and variance of a sum of N random variables where N is itself a random variable

Question

I want to find the expectation and variance of a sum of N random variables. I know that by the central limit theorem, given that N is known, the expectation should be $N\mu$ and the variance should be $N\sigma^2$ where $\mu$ and $\sigma^2$are the mean and variance of the original distribution. But how should I extend this to the case where N is a random variable from another distribution, say with $\mu_2$ and $\sigma_2^2$?

Answer 1

I think you first may want to assume $N$ comes from a discrete non-negative distribution (otherwise, it's not clear how to add up -1.5 random variables). Then use Wald's Identity for expected value and there are related results in similar style for the variances.