Standard Deviation of product of two Gaussian Distribution

Question

If we have, $Z = XY,$ where $X$ and $Y$ have Gaussian Distribtuion, and both are independent.

I solved with the Monte Carlo Algorithm, it shows some values of Standard Deviation, but I don't know what is the formula for finding the Standard deviation of product of two Gaussian Distribution ?

Answer 1

Suppose $X = N(\mu_1,\sigma^2_1), Y = N(\mu_2, \sigma^2_2)$ independent . $E[Z] = E[X]E[Y] = \mu_1\mu_2$. $E[Z^2] = E[X^2]E[Y^2] = (\sigma_1^2+\mu_1^2)(\sigma_2^2+\mu_2^2)$

$Var(Z) = (\sigma_1^2+\mu_1^2)(\sigma_2^2+\mu_2^2) - \mu_1^2\mu_2^2$

Answer 2

Hint $:$ Independence $\implies$ Uncorrelated.

Answer 3

One of the guy published a paper on this topic.

link