I was wondering if there was a formula for the expectation of a product of $n$ independent random variables. I have only seen one for two random variables.
I guess what I am asking is:
Let $X_1, \dots, X_n$ be $n$ independent random variables. Is there a formula for $E\left(\prod\limits_{i=1}^n X_i\right)$?
I need this for a problem I am working on. It would be nice if it was equal to $\prod\limits_{i=1}^n E(X_i)$. While this is true when $n=2$, I'm not sure if this is true for general $n$.
If you are able to provide a formula for this I would appreciate a source or some sort of derivation.
Best Answer
If the values are independent then, it basically follows from the definition of expectation and the fact of independence of the series (that the joint probability function is the product of the marginal probability functions):
$$\begin{align} \mathsf E(\prod_i X_i) & = \iiint (\;\prod_i x_i f_i(x_i)\;)\; \mathrm d x_n\cdots\mathrm d x_1 \\ & = \int x_n f_n(x_n)\mathrm d x_n \cdot \iiint\prod_{i=1}^{n-1}x_if_i(x_i)\mathrm d x_{n-1}\cdots\mathrm d x_1 \\ & = \prod_i \int x_i f_i(x_i)\mathrm d x_i \\ & = \prod_i \mathrm E(X_i) \end{align}$$
Or you might prefer to use the Law of Iterative Expectation: $$\begin{align} \mathsf E(\prod_{i=1}^n X_i) & = \mathsf E(\mathsf E(X_n\prod_{i=1}^{n-1} X_i\mid X_n)) \\ & = \mathsf E(X_n\mathsf E(\prod_{i=1}^{n-1} X_i)) \\ & = \mathsf E(X_n)\mathsf E(\prod_{i=1}^{n-1} X_i) \\ & = \prod_{i=1}^n \mathsf E(X_i)\end{align}$$