For any
.
So
and .
SO, let t= -2 in the MGF of X and it's over.
I've never seen this before and I was trying a lot of fancy things, but all they want you to do is recognize the definition of the MGF.
X is non-negative with mgf = (1/1-2t)e^(t/1-2t) t<1/2
Pr(Y=y|X=x) = Poisson with parameter 2X
Find Pr(Y=0)
Any hints on where to start? I applied law of total probability etc, but can't get Pr(X=x) from the mgf, or can I?
Don't necessarily want to have this exact problem solved as I want to be able to solve it myself, however looking for insight into where to start the thinking process from in order to solve similar problems.
Thanks