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My textbook, Introduction to Probability, by Blitzstein and Hwang, provides the following problem:

Let $X = V + W$ and $Y = V + Z$ where $V$, $W$, and $Z$ are independent $\text{Pois}(\lambda)$ random variables.

I want to then find the joint distribution of $X$ and $Y$ so that I can calculate $P(X = 3, Y = 0)$ in terms of $\lambda$.

However, I am confused about how to proceed here. I would greatly appreciate it if people could please take the time to show how to do this. I would really appreciate it if people could explain the reasoning behind the steps taken, so that I can understand what's going on in the solution process.

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You don't need to find the joint distribution. $$\begin{align}P(X=3,Y=0)&=P(W=3,V=0,Z=0)\\&=P(W=3)P(V=0)P(Z=0)\\&=e^{-3\lambda}\lambda^3/3!\end{align}$$

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  • $\begingroup$ Thanks for the answer. How did you get that $Y = 0$ means that $V = 0$ and $Z = 0$? Please explain your answer in more detail. $\endgroup$ – The Pointer Oct 15 '19 at 11:23
  • $\begingroup$ Poisson Rvs can't be negative, and if sum of two Poissons is $0$, both must be $0$. $\endgroup$ – gunes Oct 15 '19 at 11:23
  • $\begingroup$ Hmm, I see. So would $X$ and $Y$ be independent/conditionally independent in this case (since they have $V$ in common)? $\endgroup$ – The Pointer Oct 15 '19 at 11:29
  • $\begingroup$ $X$ and $Y$ are not independent, but conditionally independent given $V$, which is the common entry. $\endgroup$ – gunes Oct 15 '19 at 12:01

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