# Understanding the bivariate Poisson distribution [duplicate]

Researching bivariate Poisson over the web is no easy task unless you can make sense of the Greek symbols.

I am familiar with Poisson and have a deep understanding of it. So could someone explain bivariate Poisson with its formula so that I can fathom it. My whole model it seems should be based on bivariate Poisson and not Poisson.

I have found this paper to be helpful but am still struggling: "Bayesian and Non-Bayesian Analysis of Soccer Data using Bivariate Poisson Regression Models" by D. Karlis and J. Ntzoufras.

I understand that the univariate model treats each teams goal scoring as a stand-alone independent process. Thus, the probability of the home team scoring ($\text{HG}$) 1 goal is the same whether the away team scores ($\text{AG}$) 0,1,2 or 10 goals, i.e.:

$$p(\text{HG}=\alpha, \text{AG}=x) = p(\text{HG}=\alpha, \text{AG}=y) \quad \text{for all \alpha, x and y}$$

So the expression above is independent of $x$:

$$p(\text{HG}=\alpha,\text{AG}=x)=f(\alpha) \quad \text{for all x}$$

With the bivariate distribution, the probability function is a function of both the home and away scores. This allows a dependency between the home and away scores to be included in the model.

But I would appreciate it if someone gave a complete example with a working out as it will help my get a thorough understanding of bivariate Poisson.

• The defining formula is at the bottom of p. 7 in your reference. Subsequent pages describe some properties. The first one is that both marginals are Poisson: in other words, this is a pair of Poissons with some possible correlation between them given by the third parameter $\theta_0$. You can leverage your understanding of the (univariate) Poisson by first taking $\theta_0=0$ and checking that this really describes a pair of independent Poissons. The really astonishing conclusion (to me, anyway) occurs a little later: the distribution of the difference does not depend on the correlation! – whuber Jan 15 '13 at 14:39
• so say for example, I find PoissonTeam1(1 goal, 1.2 mean) and PoissonTeam2(3 goal, 2.2 mean). I guess I will need a pre-determined value for theta0 ? stupid question but i am confused by that reference pdf since I dont use theta or lambda for my formulas and its confusing me as to what is what? Would u mind going over the formula please. Pretty please. Really desperate to learn this. – O P Jan 15 '13 at 15:20
• The covariance parameter $\theta_0$ for a particular match is, just like $\theta_1$ & $\theta_2$, determined from the match predictors according to a log-linear model. The coefficients for the model are fit beforehand to a large data-set. Karlis & Nztoufras (2003), "Analysis of sports data by using bivariate Poisson models" gives a good review. – Scortchi - Reinstate Monica Jan 16 '13 at 10:15
• See the answers too stats.stackexchange.com/questions/108705/… – kjetil b halvorsen Jul 21 '14 at 15:04