Let $\displaystyle $X$$ and $\displaystyle $Y$$ be jointly normally distributed random variables with correlation $\displaystyle $0 \le \rho \le 1$$ and variance $\displaystyle $\sigma^2 = 1$$. How do I find the moment generating function for $\displaystyle $X \cdot Y$$? I first thought of doing a transformation of variables so that $\displaystyle $U=X \cdot Y $$ and $\displaystyle $V=Y \imples$$, finding the joint density $\displaystyle f_{U,V}(u,v)$ and integrating over $\displaystyle $V$$, but unfortunately this seems to lead to very messy calculations. Is there some cleaner way of dealing with this problem?