Estimate of the mean number of events per period l | Vose Software

# Estimate of the mean number of events per period l

Like the binomial probability p, the mean events per period l is a fundamental property of the stochastic system in question. It can never be observed and it can never be exactly known. However, we can become progressively more certain about its value as more data are collected. Statistics provides us with a means of quantifying the state of our knowledge as we accumulate data.

We discuss two approaches: Bayesian and classical statistics

##### 1. Bayesian inference

Assuming an uninformed prior p(l) = 1  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, Compare this to the density function for a Gamma(a,b) distribution:

which by comparison shows that the two equations are equal for a Gamma(a+1,1/t) 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(a,b) distribution, the posterior is given by a Gamma(a + a, b/ (1 + b t)) distribution.

##### 2. Classical statistics

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.