# Random Variables Expectation Product

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?

• What's $*$? And if you know the first equality holds (whatever it means) for any random variables $X$ and $Y$, why can't you just replace $Y$ by $Y^2$ to get the second equality? Apr 19, 2018 at 11:17
• Are $X$ and $Y$ independent? There are counter-examples if they are merely uncorrelated Apr 19, 2018 at 11:20
• Yes they are independent. Apr 19, 2018 at 11:21
• I believe the answer is Yes; because of linearity of expectation. Apr 19, 2018 at 11:23

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