What is the effect of the variance on a sequence of cumulative product?

Question

We randomly draw numbers from a normal distribution with mean equals $mu$ and variance equals $var$.

We draw the values: $x_1, x_2, x_3, x_4, ...$

Then, we construct a sequence made of the cumulative product of these randomly drawn numbers.

$(x_1), (x_1\cdot x_2), (x_1\cdot x_2\cdot x_3), ... $

The expected values of any number of this sequence depends not only on $mu$ but also on $var$. What is the formula for the expected value (given $mu$ and $var$) of say the 15th value of this sequence?

Answer 1

If the $X_i$s are independent, then $E(\prod_{i=1}^n X_i) = \prod_{i=1}^n E(X_i) = \mu^n$.