Solution
= Solution
For the standard <normal density>, $P'(x)=-xP(x)$. Using <integration by parts>,
$$
\mathbb E[X^4]=-\int_{-\infty}^\infty x^3P'(x)dx=-[x^3P(x)]_{-\infty}^\infty+3\int_{-\infty}^\infty x^2P(x)dx.
$$
The boundary term vanishes because <Gaussian decay dominates every power>. Since the given <mean> is zero and <variance> is one, $\mathbb E[X^2]=1$, yielding the <Gaussian fourth moment>
$$
\boxed{\mathbb E[X^4]=3.}
$$