Compute mean and variance.

Question

Let $X$ be a random variable that is normally distribted with mean $\mu$ and variance $\sigma^2$. Compute

1. $E(X^2)$
2. $E(X^4)$
3. $Var(X^2)$

They also hint that for a standard normally distributed random variable $Z$ it follows that $E(Z^4)=3$.

I don't see how to use the hint given

Answer 1

$\newcommand{\e}{\operatorname{E}}$\begin{align} \e(X^4) & = \e((\mu+\sigma Z)^4) \\[10pt] & = \e(\mu^4 + 4\mu^3\sigma Z + 6\mu^2\sigma^2 Z^2 + 4 \mu\sigma^3 Z^3 + \sigma^4 Z^4) \\[10pt] & = \mu^4 + 4\mu^3\sigma \e(Z) + 6\mu^2\sigma^2 \e(Z^2) + 4\mu \sigma^3 \e(Z^3) + \sigma^4 \e(Z^4). \end{align} Now apply what you were given about the expected values of powers of $Z.$