# Thread: Joint probability function

1. ## Joint probability function

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

2. For any $y=0,1,2...$

$P(Y=y) =\sum_{x=0}^{\infty}P(Y=y|X=x)P(X=x)$.

So $P(Y=y) =\sum_{x=0}^{\infty}{e^{-2x}(2x)^y\over y!}P(X=x)$

and $P(Y=0) =\sum_{x=0}^{\infty}e^{-2x}P(X=x) =MGF_X(-2)$.

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.