Expectation of product of 3 normal distributed variables

Question

Given are 3 independent normal variables $X_i=\mathcal{N}(\mu_i,\sigma^2)_{i=1,2,3}$ with expectations $\mu_i$ and equal variance $\sigma^2$.

What is the expectation of their product $\mathbb{E}[X_1 \cdot X_2 \cdot X_3]$ ?

Answer 1

Do the separate integrals to find:

$${\cal E}[X_1 X_2 X_3]$$

$$= {\cal E}[X_1]{\cal E}[X_2]{\cal E}[X_3]$$

$$= \underbrace{\left( \int\limits_{x=-\infty}^\infty \frac{x}{\sqrt{2 \pi} \sigma} e^{-(x - \mu_1)^2/(2 \sigma^2)}\ dx \right)}_{\mu_1} \underbrace{\left( \int\limits_{y=-\infty}^\infty \frac{y}{\sqrt{2 \pi} \sigma} e^{-(y - \mu_2)^2/(2 \sigma^2)}\ dx \right)}_{\mu_2} \underbrace{\left( \int\limits_{x=-\infty}^\infty \frac{z}{\sqrt{2 \pi} \sigma} e^{-(z - \mu_3)^2/(2 \sigma^2)}\ dx \right)}_{\mu_3}$$

$$= \mu_1 \mu_2 \mu_3,$$

independent of the variances. Note that the three variances could differ from each other and you'd get the same result.