# Poisson is to exponential as Gamma-Poisson is to what?

A Poisson distribution can measure events per unit time, and the parameter is $\lambda$.
The exponential distribution measures the time until next event, with the parameter $\frac{1}{\lambda}$.
One can convert one distribution into the other, depending on whether it is easier to model events or times.

Now, a gamma-poisson is a “stretched” poisson with a larger variance. A Weibull distribution is a “stretched” exponential with a larger variance. But can these two be easily converted into each other, in the same way Poisson can be converted into exponential?

Or is there some other distribution that is more appropriate to use in combination with the gamma-poisson distribution?

The gamma-poisson is also known as the negative binomial distribution, or NBD.

This is a fairly straight forward problem. Although there is a connection between the Poisson and Negative Binomial distributions, I actually think this is unhelpful for your specific question as it encourages people to think of negative binomial processes. Basically, you have a series of Poisson processes:

Where $Y_i$ is the process and $t_i$ is the time you observe it, and $i$ denotes the individuals. And you are saying that these processes are “similar” by tying the rates together by a distribution:

On doing the integration/mxixing over $\lambda_i$, you have:

This has a pmf of:

To get the waiting time distribution we note that:

Differentiate this and you have the PDF:

This is a member of the generalized Pareto distributions, type II. I would use this as your waiting time distribution.

To see the connection with the Poisson distribution, note that $\frac{\alpha}{\beta}=E(\lambda_i|\alpha\beta)$, so that if we set $\beta=\frac{\alpha}{\lambda}$ and then take the limit $\alpha\to\infty$ we get:

This means that you can interpret $\frac{1}{\alpha}$ as an over-dispersion parameter.