I have X,Y ~ i.i.d. Normal(M, S), where in turn M ~ N(2,1) and S ~ exponential(1).
I have to evaluate COV(X,Y) and E(X^2 Y^2).
For COV(X,Y), I need E(X) = M, E(Y) = M. I also need E(XY), which I calculated using the usual method:
E(XY) = double integral (-inf to inf) x*y * (pdf of X) (pdf of Y) dx dy
Here PDF of X and PDF of Y turn out to be functions of M and S, and the double integral evaluated by parts leads me to an answer which gives (inf - (-inf)), which doesn't make sense.
I think I should probably use some other method to work on distributions which have random variables as parameters. I checked the textbook Intro to Probability models by Sheldon Ross but couldn't find anything on compound random variables.
I'll be grateful for any guidance on how to proceed here... Thanks a lot.
I assume the hierarchical model is:
$$M \sim \operatorname{Normal}(\mu = 2, \sigma = 1), \\ S \sim \operatorname{Exponential}(1), \\ X \mid M, S \sim \operatorname{Normal}(\mu = M, \sigma = S), \\ Y \mid M, S \sim \operatorname{Normal}(\mu = M, \sigma = S).$$
Note that I am assuming that $S$ is the conditional standard deviation of $X$, rather than the variance. Then we have $$\begin{align} \operatorname{E}[X] &= \operatorname{E}[\operatorname{E}[X \mid M, S]] \\ &= \operatorname{E}[M] \\ &= 2, \end{align}$$ and similarly for $Y$. And
$$\begin{align} \operatorname{Var}[X] &= \operatorname{E}[\operatorname{Var}[X \mid M, S]] + \operatorname{Var}[\operatorname{E}[X \mid M, S]] \\ &= \operatorname{E}[S^2] + \operatorname{Var}[M] \\ &= \operatorname{Var}[S] + \operatorname{E}[S]^2 + \operatorname{Var}[M] \\ &= 1^2 + 1^2 + 1^2 = 3. \end{align}$$
The unconditional expectation of $XY$ is
$$\begin{align} \operatorname{E}[XY] &= \operatorname{E}[\operatorname{E}[XY \mid M, S]] \\ &= \operatorname{E}[M^2] \\ &= \operatorname{Var}[M] + \operatorname{E}[M]^2 \\ &= 1^2 + 2^2 = 5. \end{align}$$
From this, it is easy to compute the covariance. I leave the calculation of $\operatorname{E}[X^2 Y^2]$ as an exercise. The relevant formulas we have used here are the laws of total expectation and total variance.