Name of theorem for $\mathbb{E}\left[ \sum_{n=1}^N X_n \right] = \mathbb{E}[N] \mathbb{E}[X_1]$

Question

Suppose we have identically distributed variables $(X_n)_n$ and some random variable $N$ which takes values in $\mathbb{N}$. Then under certain conditions we have: $$ \mathbb{E}\left[ \sum_{n=1}^N X_n \right] = \mathbb{E}[N] \mathbb{E}[X_1], $$ but where can I find this theorem and does it have a name?

Answer 1

The random variable $\sum_{n=1}^{N}X_{n}$ is called a compound random variable. It is the sum of a random number of iid random variables. In particular, if the random variable $N$ is Poisson distributed, then the random variable $\sum_{n=1}^{N}X_{n}$ is called a compound Poisson random variable.