X and Y have the following joint moment generating function:
$M_{X,Y}(a,b) = \Large \frac{4}{5}[\frac{1}{(1-a)(1-b)}+\frac{1}{(2-a)(2-b)}]$
Find E(XY)
I have gone through this problem several times, first calculating the derivative with respect to a and then subsequently calculating the derivative with respect to b.
I think I can spot an exponential distribution, but I'm not sure how to make the computation easier.
Edit: I have a final answer of .85. Not sure if this is correct or not.
Naturally, you can take derivative $\partial^2_{a,b} M_{X,Y}(0,0)$ to get the answer. Alternatively, you can interpret the joint distribution of $X$ and $Y$ as a mixture. Namely, a random variable $\Lambda$ takes value $1$ with probability $4/5$ and value $2$ with probability $1/5$. Given $\Lambda = \lambda$, $X$ and $Y$ are independent and exponentially distributed with parameter $\lambda$. Then $$ E[XY] = E[E[XY|\Lambda]] = E[\Lambda^{-2}] = 1\cdot\frac{4}{5} + \frac{1}{4}\cdot\frac{1}{5} = 0.85, $$ confirming your finding.