# Find three Poisson-distributed random variables, pairwise independent but not mutually independent

I am asked to give an example of three Poisson-distributed random variables which are pairwise independent, but are not mutually independent.

I thought of the example about the Intersection where cars are coming from one side and they can go to one of three directions left, right or keep straight, with probabilities $p \$, $q \$ and $1-p-q$, respectively.

However, I find it hard to prove.

Let $p(x,y,z)$ be the joint PMF for independent Poisson random variables with parameters $\lambda_1, \lambda_2, \lambda_3$. I will modify this slightly to obtain a joint PMF $q(x,y,z)$. The requirement for the marginals $X,Y,Z$ to have the correct distributions and be pairwise independent is that $$P(X=x,Y=y) = \sum_{z=0}^\infty q(x,y,z) = \sum_{z=0}^\infty p(x,y,z)$$ and similarly for the sums over $x$ and $y$. Of course we need all $q(x,y,z)$ to be nonnegative. We can manage this by starting with $q = p$ and slightly changing $8$ of the entries: add $\epsilon$ to $q(0,0,1), q(0,1,0),q(1,0,0)$ and $q(1,1,1)$ and subtract $\epsilon$ from $q(0,0,0), q(0,1,1), q(1,0,1)$ and $q(1,1,0)$, where $\epsilon$ is small enough that all of these are still nonnegative.
• I have two more questions: (1) the random variables $X,Y,Z,\ldots$ etc don't have to be non-negative, right? I don't see a reason why they should. (2) This system procedure is applicable to a large class of continuous distributions as well, isn't it? Just recently there was this post (math.stackexchange.com/questions/2029257) with isolated ideas of different flavors, and maybe this method can be modified to appear over there as well? – Lee David Chung Lin Dec 8 '16 at 5:30