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.
Of course there is nothing very special about the Poisson distributions here: you could do something similar for any nontrivial discrete distributions.
