Suppose that ~ where ~ . Find the variance and forth cumulant of Y.

It looks like use of moment generating function is appropriate here - if I find I can then calculate and and 4th cumulant. But I cannot move beyond:

(Since 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.