Mean/ Expected Value of $X^4$

Question

$\def\Var{\mathop{\rm Var}}$Can anyone help me prove that Expected Value of $X^4$ is $3\Var(X)^4$, if the Expected Value of $X$ is zero and $\Var(X)$ is the Variance of $X$?

Thanks!

Answer 1

That is wrong in general. Suppose for $\def\R{\mathbb R}\lambda \in \R^+$ we have $\def\P{\mathbb P}\P(X=\lambda) = \P(X=-\lambda) =\frac 12$, then for any $k$ we have $$ \def\E{\mathbb E}\E(X^k) = \frac 12\bigl(\lambda^k + (-\lambda)^k\bigr) = \begin{cases} \lambda^k & k \text{ even}\\ 0 & k \text{ odd}\end{cases} $$ Hence $\E(X) = 0$, ${\rm Var}(X) = \E(X^2) = \lambda^2$ and $\E(X^4) = {\rm Var}(X)^2$. So we have $\E(X^4) = 3{\rm Var}(X)^4$ only if $\lambda^4 = 3\lambda^8$, which is wrong in general.