# How do I generate two correlated Poisson random variables?

How would I simulate observations from a bivariate Poisson distribution such that they have a nonzero covariance? The hint I was given is that I need to use the fact that the sum of two Poisson random variables is also Poisson.

• See stats.stackexchange.com/questions/108705. Both answers there show how values can be simulated by means of three independent Poisson variates (although they do not explicitly point that out). – whuber Feb 21 '15 at 20:25
• Please see the self-study tag wiki. The hint you already have is a complete giveaway; I'm not sure there's much of anything between the hint you already have and telling you exactly which Poissons to add ... i.e. a complete solution. Oh, wait, there's one -- start with more than two independent Poissons. – Glen_b -Reinstate Monica Feb 22 '15 at 9:53

1. take a bivariate normal $$(x_1,x_2)\sim\mathcal{N}_2\left((0,0),\left[\matrix{1 &\rho\\\rho &1}\right]\right)$$generation;
2. turn $(x_1,x_2)$ in correlated uniforms as $$(u_1,u_2)=(\Phi(x_1),\Phi(x_2))$$where $\Phi(\cdot)$ is the normal CDF;
3. derive two Poisson variates $(n_1,n_2)$ with parameters $\lambda_1$ and $\lambda_2$ from $(u_1,u_2)$ by inverting the Poisson CDF.