I'm trying to find the covariance between two variables that are dependent, but conditionally independent. My two random variables, $X_1$ and $X_2$ are i.d. and their probability density functions are $f(x_i)=\frac{1}{\sigma\sqrt{2\pi}}e^-\frac{(x_i-M)^2}{2\sigma^2}$. But M is a random variable itself. How do I calculate $Cov(X_i,X_j)$?
Eventually, I'll want to mess around with the probability density function of M, but it can be either discrete or continuous. If it helps to answer, two simple cases could be (1) $f(M=\mu)=\frac{1}{c_2-c_1}$ on $(c_1,c_2)$ or (2) $P(M=c_1)=0.6$ and $P(M=c_2)=0.4$.
Thanks in advance for any help!
Note that $X_1=M+\sigma Y_1$ and $X_2=M+\sigma Y_2$ where $(Y_1,Y_2)$ is i.i.d. standard normal and independent of $M$, hence $E[X_i]=E[M]$, $X_1X_2=M^2+\sigma M(Y_1+Y_2)+\sigma^2Y_1Y_2$ and $E[X_1X_2]=E[M^2]$, in particular $\mathrm{Cov}(X_1,X_2)=\mathrm{var}(M)$.