$E[\exp(XY)]$=? $X=Z+\epsilon_1$, $Y=Z+\epsilon_2$, $Z, \epsilon_1, \epsilon_2\sim N(0,1)$, i.i.d.

Question

Suppose $Z, \epsilon_1, \epsilon_2\sim N(0,1)$, i.i.d. Define $X=Z+\epsilon_1$, $Y=Z+\epsilon_2$. Want to find $E[\exp(XY)]$.

Does anyone have a hint on this? The hard part is how to deal with the exponential function...

Answer 1

Hint:

(1) You can always do a brute force integration. Read this post if you are still stuck.

(2) The law of total expectation can be useful. Try to calculate conditional expectation of $X$ given $Y$ first. You will need to use MGF of normal distribution.

Answer 2

Hint

$$\Large E=\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty {1\over \sqrt {8\pi^3 }}e^{z^2+(\epsilon_1+\epsilon_2)z+\epsilon_1\epsilon_2}e^{-{z^2\over 2}}e^{-{\epsilon_1^2\over 2}}e^{-{\epsilon_2^2\over 2}}dz d\epsilon_1d\epsilon_2$$

and use the following $$\int_{-\infty}^{\infty}e^{-{(x-\mu)^2\over 2\sigma^2}}=\sigma\sqrt{2\pi}$$