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Home > Probability theory and statistics > Stochastic processes > The Poisson process > Estimate of the mean number of events per period lambda

# Estimate of the mean number of events per period l

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We discuss two approaches: Bayesian and classical statistics

#### 1. Bayesian inference

Assuming an uninformed prior p(l) = 1/ l  and the Poisson likelihood function for observing a events in period t:

The proportional statement is acceptable because we can ignore terms that don't involve l, and we then get the posterior distribution:

which by comparison with a Gamma density function is a Gamma(0,1/t,a) distribution. The Gamma distribution can also be used to describe our uncertainty about l if we start off with an informed opinion and then observe a events in time t. If we can reasonably describe our prior belief with a Gamma(0,b,a) distribution, the posterior is given by a Gamma(0, b/ (1 + b t),a + a) distribution.

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Various classic statistics approaches to estimating l are discussed here.

#### 3. Comparison of classical and Bayesian methods

A comparison of the different approaches to estimating l are discussed here.