What is the expected value of $E[X^2 Y]$ if X and Y are independent? Given X, Y independent, how can we compute $E[X^2 Y]$?
 A: The short answer is that $$E(X^2 Y) = E(X^2) E(Y)$$ as independence is preserved under transformations. In general, if $X$ and $Y$ are independent, then $f(X)$ and $g(Y)$ will be independent.
Note however that this does not simply any further. We cannot say that $E(X^2) E(Y) = E(X)^2 E(Y)$ as this is untrue in general.
A: $$\begin{align}\mathsf E(X^2Y)&=\mathsf E(\mathsf E(X^2Y\mid Y))&&\text{Law of Total Expectation}\\&=\mathsf E(\mathsf E(X^2\mid Y)\cdot Y)&&Y\text{ relatively constant to a conditional expectation wrt }Y\\&=\mathsf E(\mathsf E(X^2)\cdot Y)&&\bigstar\\&=\mathsf E(X^2)\cdot \mathsf E(Y)&&\mathsf E(X^2)\text{ is a constant}\\[3ex]&=\big(\mathsf{Var}(X)+\mathsf E(X)^2\big)\cdot \mathsf E(Y)&&\text{by definition for Variance.}\end{align}$$
$^\star:$ $X$ and $Y$ are independent, so the realised value of $Y$ has no influence on the expected value of $X$ nor any monovariate function of $X$ (such as $X^2$).   Therefore the conditional expectation with respect to $Y$ is just the expectation.
A: $X$ and $Y$ are independent $\Rightarrow X^{2}$ and $Y$ are independent.
Let us recall some definitions. Let $(\Omega,\mathcal{F},P)$ be a
probability space. Let $\mathcal{G}_{1},\mathcal{G}_{2}\subseteq\mathcal{F}$.
We say that $\mathcal{G}_{1},\mathcal{G}_{2}$ are independent if
$P(A\cap B)=P(A)P(B)$ for any $A\in\mathcal{G}_{1}$ and $B\in\mathcal{G}_{2}$.
Now, if $X$ and $Y$ are random variables, we say that $X$ and $Y$
are independent if $\sigma(X)$ and $\sigma(Y)$ are independent.
Note that $\sigma(X^{2})\subseteq\sigma(X)$ (because $X^{2}$ is
$\sigma(X)/\mathcal{B}(\mathbb{R})$-measurable and $\sigma(X^{2})$
is the smallest $\sigma$-algebra on $\Omega$ such that $X^{2}$
is measurable). Now, it is clear that $\sigma(X)$ and $\sigma(Y)$
are independent $\Rightarrow$ $\sigma(X^{2})$ and $\sigma(Y)$ are
independent.
