Let $X, Y$ and $Z$ be three independent random variables such that $E(X)=E(Y)=E(Z)=0$ and $Var(X)=Var(Y)=Var(Z)=1$. Calculate $E[(X^2)(Y+5Z)^2]$
I know that the answer is $26$.
Since all of the expected values of $X, Y$ and $Z$ are all the same, I have replaced each expected value of $X, Y$ and $Z$ with just $E[X]$. For example, $E[(X^2)(Y^2)]$ is now $(E[X])^4$.
Doing this, I'm left with $26E[X^4]$.
Since the variance of each random variable is one, I know I need to somehow turn $E[X^4]$ into the formula for variance.... Thanks guys