Csdrhrt

So Wald's Equation states that for a real-values, independent and identically distributed sequence of random variables $(X_{n})_{ninmathbb{N}}$ and a nonnegative integer $N$, which is independent of the sequence, we have that:

$mathbb{E}[X_{1}+dots+X_{N}]=mathbb{E}[N]mathbb{E}[X_{1}]$

Under the assumption that both have finite expectation.

My question is if there exists some extension or similar identity that can be applied to the same type of problem except then with multiplication. I'm not sure if this simply follows from independence in most cases though. For example, given the same setting as earlier, do we have that:

$mathbb{E}[prod^{N}_{i=1}X_{i}]=prod^{N}_{i=1}mathbb{E}[X_{i}]$

Or is the outcome something comparable to the Wald's Equation?

Any help is appreciated!

Let $ N, X_1, X_2, dots $ independent random variables, $ (X_i) $ identically distributed and integrable with $ mu = mathbb{E}[X_1] $, and $N$ take integer values with $ mathbb{E}[{(mathbb{E}|X_1|)}^N]$ well defined. Then
$$ mathbb{E}left[Pi_{i=1}^NX_iright]=mathbb{E}[mu^N].$$

In fact,
$$mathbb{E}left[Pi_{i=1}^NX_iright]=mathbb{E}left[sum_{n=1}^infty 1_{{N=n}}Pi_{i=1}^N X_iright]= sum_{n=1}^infty mathbb{E}left[1_{{N=n}}Pi_{i=1}^N X_iright]=sum_{n=1}^infty mathbb{E}left[1_{{N=n}}Pi_{i=1}^n X_iright]=sum_{n=1}^infty {mathbb{E}[X_1]}^n mathbb{P}(N=n)
=mathbb{E}[mu^N].$$
Note that in the penultimate equality we require independence and in the last equality we need a hypothesis about $ mu^N $. Initially work with $$|Pi_{i=1}^N X_i|$$
instead of $$Pi_{i=1}^N X_i$$ to show the integratibility.