1. ## Moment generating function

Consider a hierarchical model that defines the joint distribution of (X,Z). Let
X|Z=z ~ N(0,z^2) where Z ~ N(0,1). Determine the moment generating function of X.

My solution so far is attached. Am I going in the right direction? If so, how do I solve the integral for the marginal PDF of X? Or am I doing completely useless stuff?

Thanks

2. 1 Your attachment has errors in it
2 I also started working on this IGNORING the MGF comment
3 THEN I realized that z is your standard deviation, hence it cannot be a N(0,1) rv, try a chi-square maybe. Or U(0,1)?