Suppose that that random variable $N$ follows a Poisson distribution with mean $\lambda=6$. Suppose that the conditional distribution of the random variable $X$, given that $N=n$, follows that of a $Binomial(n,0.6)$.
Find the Joint Moment Generating Function of $(N, X)$.
Initially I just tried to use the definition. I found the joint PMF using the definition of the conditional distribution, but then I have to sum over both of them in order to find the joint MGF, and this was the step I was stuck at because trying to do a double sum over the product of the binomial and Poisson PMFs doesn't exactly go over very nicely.
Since this problem does NOT assume independence, I can't exactly attempt to use that to my advantage either... so now I'm stuck. How can I calculate this joint MGF?