From my recent question here: Calculating the probability of a car accident
I wish to learn how to apply a multivariate poisson distribution in the example that car-accidents vary for each year, therefore this model will require in effect to take into account this variation.
A simple multivariate Poisson model looks like this:
The joint probability function is given by \begin{align} P(X)&=P(X_1=x_,X_2=x_2,\ldots,X_m=x_m) \\&=\exp\left(-\sum_{i=1}^m \theta_i\right)\prod_{i=1}^m \frac{\theta_i^{x_i}}{x_i!}\sum_{i=0}^s \prod_{j=1}^m \binom{x_j}{i}i!\left(\frac{\theta_0}{\prod_{i=1}^m \theta_i}\right)^i\,, \end{align}
where $s=\min(x_1,x_2,\ldots,x_m).$
(taken from: http://www2.stat-athens.aueb.gr/~karlis/multivariate%20Poisson%20models.pdf)
Though I'm unfamiliar with the equation such as why is the formula on the right of the poisson distribution introduced the way that it is? And what's an example of it's use (preferably an R example)
