Random Variables Expectation Product

Question

Let $X$ and $Y$ are independent random variables. Can I say that $E[XY^2] = E[X]*E[Y^2]$? I know that $E[XY] = E[X]*E[Y]$ but does this still hold true if a random variable is squared?

Answer 1

\begin{align}E(XY^2) &= \int \int xy^2 P(x,y) \, dx \, dy \\ &= \int \int xy^2 P(x) P(y) \, dx \, dy & \text{ (as the they are independent)}\\ & = \int x P(x) dx\int y^2 P(y)dy \\ & = E(X) E(Y^2) \end{align}

Answer 2

Theorem: If $X_1, X_2,...,X_n$ are independent random variables and for $i=1,2,...,n,$ the expectation $\mathbb{E}[f_i(X_i)]$ exists then:$$\mathbb{E}\left[\prod_{i=1}^{n}f_i(X_i)\right]=\prod_{i=1}^{n}\mathbb{E}[f_i(X_i)].$$ Proof is similar to @kasa's answer. It can be proved more general:

Theorem:$X_1, X_2,...,X_n$ are mutually independent $\Longleftrightarrow$ $$\mathbb{E}\left[\prod_{i=1}^{n}f_i(X_i)\right]=\prod_{i=1}^{n}\mathbb{E}[f_i(X_i)]$$ for eny $n$ function $f_i$ such that the expected values exist and are well-defined