Overview
A Poisson random variable is a variable whose output can be a Natural number, that is it can be any of the following {% 0,1,2,3,4 ... %}.
As such, a Poisson variable is often used to represent count data, that is, it represents the probability of an event occurring a given number of times. (presumably, within some given time frame) The Poisson variable is modeled using the Poisson Distribution.
The Poisson distribution can be used in the following ways.
- Modeling the number of times a claim occurs for a single policy. This is done when the policy can have more than one claim over time, such as auto insurance.
- Probability of a single claim on a single policy. While not presumably a perfect fit, when the probability of a claim occurring is low, the Poisson variable can be interpreted such that any value greater than or equal to one is interpreted to mean the claim has occurred.
Fitting to Data
The moments of the Poisson distribution are given by
{% \displaystyle \mathbb{E}[x] = \lambda %}
{% \sigma^2 = \lambda %}
This mean that the value of {% \lambda %} can be estimated by simply averaging the counts in the observed
dataset.
{% \sigma^2 = \lambda %}
Factors Influencing Probability
Often it is necessary to consider to effect of exogenous factors on the probability specified in the Poisson model. This is typically done using a Poisson Regression