Suppose that $\displaystyle Y|Z=\sigma^2{$~$\displaystyle N(0,\sigma^2)$ where $\displaystyle Z{$~$\displaystyle }Exp(\lambda)$. Find the variance and forth cumulant of Y.

It looks like use of moment generating function is appropriate here - if I find $\displaystyle M_Y(t)$ I can then calculate $\displaystyle Var(Y)$ and $\displaystyle K_Y(t)$ and 4th cumulant. But I cannot move beyond:

$\displaystyle M_Y(t)=E[M_{Y|Z}(y|z)]=E[exp(\frac{1}{2}t^2{\sigma}^2)]=$

(Since $\displaystyle f_{Y|Z}(y|z)$ is normally distributed with mean=0)

Am I on the wrong path? What also worries me that I am not using Z distribution here at all.

So, as alternative, I was thinking going the 'long' route:

- find joint density

- then find marginal density of Y

- then find Var(Y), My(t) etc the 'long' way.