Solution
= Solution
The standard <normal density> is <even>, so $x^3P(x)$ is <odd>. Since <Gaussian decay dominates every power>, the integrand is absolutely integrable. Therefore
$$
\boxed{\mathbb E[X^3]=\int_{-\infty}^\infty x^3P(x)dx=0.}
$$
= Solution
The standard <normal density> is <even>, so $x^3P(x)$ is <odd>. Since <Gaussian decay dominates every power>, the integrand is absolutely integrable. Therefore
$$
\boxed{\mathbb E[X^3]=\int_{-\infty}^\infty x^3P(x)dx=0.}
$$