# Finding joint distribution of sum of independent Poisson random variables and calculating probabilities [duplicate]

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.

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}
• 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. – The Pointer Oct 15 '19 at 11:23
• Poisson Rvs can't be negative, and if sum of two Poissons is $0$, both must be $0$. – gunes Oct 15 '19 at 11:23
• Hmm, I see. So would $X$ and $Y$ be independent/conditionally independent in this case (since they have $V$ in common)? – The Pointer Oct 15 '19 at 11:29
• $X$ and $Y$ are not independent, but conditionally independent given $V$, which is the common entry. – gunes Oct 15 '19 at 12:01