Q: Suppose random variables and have joint moment generating function:
What are the distributions for X and Y and their covariance, Cov[X,Y]?
I am unsure how to derive the individual distributions when given just the moment generating function. Also, I am unsure how to derive the individual distributions when it is not known whether X and Y are independent/dependent random variables.
Would really appreciate it if someone can provide an answer to the question and how to evaluate the covariance. Thank you for your help in advance.
When $M_{X,Y} (s,t)$ splits as a product of a function of $s$ and a function of $t$ it follows that $X$and $Y$ are independent. Also, $M_X (s)=\frac 1 {2-s}$ and $M_Y(t)=(\frac 2 3 )^{4}(\frac 3 {3-t})^{4}$. By independence the covariance is $0$. I will let you write down the distributions of $X$ and $Y$.