I'm trying to calculate the expected value and variance of the product of n (a known integer) iid Normal Random variables with no luck!
$X_i \sim \mathcal{N}(\mu,\,\sigma^2), $
$Z = \prod_{i=1}^n X_i $
Since $X_i$ are independent, I used the product rule and wrote
$ E[Y] = E[X_1] \times ... \times E[X_n] = \mu \times ... \times \mu = \mu^n $
For Variance, I used the definition:
$ Var(Y) = E[Y^2] - (E[Y])^2 = E[X_1^2 \times ... \times X_n^2] - (E[X_1 \times ... \times X_n])^2 $
Calculating the value for $(E[X_1 \times ... \times X_n])^2$ is easy; $\mu^{2n}$
I can't get the value for $E[Y^2]$. I used the definition of expected value and get to an integral that is diverging.
$E[Y^2] = E[X_1^2 \times ... \times X_n^2] = \idotsint_{-\infty}^\infty c_1x_1^2e^{1/2({x_1-\mu}/\sigma)^2}...c_nx_n^2e^{1/2({x_n-\mu}/\sigma)^2} d_{x_1}...d_{x_n} $ and $c_i$ is the coefficient for PDF of the normal distribution. I can take the $c_i$ out, so it would be this integral:
$ c_1...c_n\idotsint_{-\infty}^\infty x_1^2e^{1/2({x_1-\mu}/\sigma)^2} ...x_n^2e^{1/2({x_n-\mu}/\sigma)^2} d_{x_1}...d_{x_n} $
As noted in comments $X_i^2$ are independent if $X_i$ are
$E(X_1^2\dots X_n^2 )=\prod _{i =1}^n E(X_1^2)= \prod _{i =1}^n \left (Var (X_1)+( EX_1)^2\right)=(\sigma^2+\mu^2)^n$
Hence the overall variance is $$(\sigma^2+\mu^2)^n-\mu^{2n}$$