Bayesian Statistics: An Introduction, 4th Edition

3.4 The Poisson distribution

3.4.1 Conjugate prior

A discrete random variable x is said to have a Poisson distribution of mean λ if it has the density

This distribution often occurs as a limiting case of the binomial distribution as the index and the parameter but their product (see Exercise 6 in Chapter 1). It is thus a useful model for rare events, such as the number of radioactive decays in a fixed time interval, when we can split the interval into an arbitrarily large number of sub-intervals in any of which a particle might decay, although the probability of a decay in any particular sub-interval is small (though constant).

Suppose that you have n observations from such a distribution, so that the likelihood is

where T is the sufficient statistic

We have already seen in Section 2.10 ...