Discrete Distributions

# The Poisson Distribution

Random variables with a Poisson distribution are generated from a process that differs slightly from that of a Binomial distribution. A Poisson random variable is a count over space or time. It is frequently used to model arrivals and departures.

## Parameters

The Poisson distribution has just one parameter. That parameter is λ (lambda), the average rate. Some sources have this parameter as μ (mu). The reparameterization is merely a substitution. Using μ emphasizes that this average rate is also the expected value. Using λ emphasizes the connection between the Poisson distribution and the Exponential distribution. The Poisson distribution can also be thought of as a count of arrivals, where the time between arrivals follows an Exponential distribution.)

Note that the sample space is defined as all non-negative integers.

## Practice

Please select the aspect of the Poisson distribution you would like to work with:

## Poisson Example

Here is a graphic of a sample Poisson distribution:

...

In this example,

X ~ Pois(λ = 4.33)

The sample space is listed across the bottom, S = {0, 1, 2, … 12}. The height of each bar represents the probability of that elementary event.