Find the moment generating function of two dependent Poisson random variables.

Question

I have a Poisson process with rate $\lambda=2$ per hour.

Let A = number of events between 6:00 PM and 9:00 PM

Let B = number of events between 8:00 PM and 10:00 PM

I need to find the moment generating function of W = A + B. Since A and be are not independent, I cannot just simply product the individual mgfs of A and B. I was thinking, I could find the pdf of W first and then use that to find the mgf, but I'm not sure where to start in order to do that.

Answer 1

$A$ and $B$ are dependent, but you can decompose them as follows:

• let $X$ be the number of events between 6:00 and 8:00
• let $Y$ be the number of events between 8:00 and 9:00
• let $Z$ be the number of events between 9:00 and 10:00

Then $A = X+Y, B = Y+Z$, and $X, Y$ and $Z$ are all independent.