Covariance of two random variables (one is squared)

Question

I have problem figuring out the solution for this task:

X1 and X2 are independent random variables with normal distribution ~N(2,1). What is a covariance of $X_1 − 4X_2^2$ and $X_1 + X_2$.

So far I've managed to come up with this:

$cov(X_1 − 4X_2^2$ , $X_1 + X_2) = cov(X_1, X_1) - cov(X_1,X_2)-4cov(X_1, X_2^2) - 4cov(X_2, X_2^2) = Var(X_1) + 0 -4cov(X_1, X_2^2) - 4cov(X_2, X_2^2)$

But I don't know how to handle both $cov(X_1, X_2^2)$ and $cov(X_2, X_2^2)$. X1 and X2 are independent, but does it mean that $X_1$ and $X_2^2$ are independent too? What about $X_2$ and $X_2^2$?

Answer 1

\begin{align} cov(X_1,X_2^2)&=EX_1X_2^2-EX_1EX_2^2 \mbox{ (by definition)}\\ &=EX_1EX_2^2-EX_1EX_2^2 \mbox{ (by independence)}\\ &=0 \end{align}

\begin{align} cov(X_2,X_2^2)&=EX_2X_2^2-EX_2EX_2^2 \mbox{ (by definition)}\\ &=EX_2^3-EX_2EX_2^2\\ &=14-2\times5\\ &=4 \end{align}

Note if $X\sim N(\mu,\sigma^2)$ then \begin{align} EX^3&=\mu^3+3\mu\sigma^2\\ EX^2&=\mu^2+\sigma^2 \end{align}