Expected value of product of three dependent normal variables

Question

Assume $X_1,X_2,X_3 \sim N(0,\sigma_i)$

Is it true that $E[X_1X_2X_3] = 0$ ? (not necessary independent variables)

Answer 1

For a specific example where this does not happen, let $X_1$ and $X_2$ be independent $\operatorname{Normal}(0,1)$ random variables, and let $$X_3 = \begin{cases} X_2 & X_1 \ge 0, \\ -X_2 & X_1 < 0.\end{cases}$$ This still leaves $X_3$ as a standard normal variable, since the normal distribution is symmetric.

In this case, $\mathbb E[X_1 X_2 X_3] = \mathbb E[|X_1| X_2 X_2] = \mathbb E[|X_1|] \mathbb E[X_2^2]$. The expected value of $|X_1|$ is $\sqrt{\frac2\pi}$, the expectation of a half-normal distribution, and the expected value of $X_2^2$ is $1$, the variance of a normal distribution. So $\mathbb E[X_1 X_2 X_3] = \sqrt{\frac2\pi} \ne 0$.