Joint Moment Generating Function Help

Question

I've been working on this problem for a while and need some direction.

$$ f(x,y) = \left\{ \begin{array}{lr} \frac{1}{\sqrt{2\pi}} e^{-x} e^{-\frac{(y-x)^2}{2}} & x \geq 0, -\infty < y < \infty, \\ 0&\mbox{otherwise}\ \end{array} \right. $$

Calculate the joint moment generating function.

My textbook hasn't been much help, but I believe the route that I should be taking involves this calculation:

$$ M_{X,Y} (t_1, t_2) = E[\exp(t_1X+t_2Y)] $$ $$...$$ $$ = E_X(\exp(t_1X)E[\exp(t_2Y)|X]) $$

After some research I've found that it relates to the "erf" function, but this was not discussed in class or in the textbook thus I believe that a different approach is required. Thanks for your help.

Answer 1

$$M_{X,Y}(t_1, t_2) = {\rm E}[e^{t_1 X + t_2 Y}] = \int_{x=0}^\infty \int_{y=-\infty}^\infty \frac{1}{\sqrt{2\pi}} e^{t_1 x + t_2 y} e^{-x} e^{-(y-x)^2/2} \, dy \, dx.$$

Answer 2

Conditioned on the value of $X$ being $x$, the distribution of $Y$ is a unit-variance normal distribution with mean $x$. So you can calculate (or look up in your text) the (conditional) moment-generating function of $Y$. So, now you have to calculate the expectation of something that looks like $e^{t_1X + t_2X + t_2^2/2}$