Documentation

## Poisson Distribution

### Definition

The Poisson pdf is

`$f\left(x|\lambda \right)=\frac{{\lambda }^{x}}{x!}{e}^{-\lambda }\text{\hspace{0.17em}};\text{\hspace{0.17em}}x=0,1,2,\dots ,\infty \text{\hspace{0.17em}}.$`

### Background

The Poisson distribution is appropriate for applications that involve counting the number of times a random event occurs in a given amount of time, distance, area, etc. Sample applications that involve Poisson distributions include the number of Geiger counter clicks per second, the number of people walking into a store in an hour, and the number of flaws per 1000 feet of video tape.

The Poisson distribution is a one-parameter discrete distribution that takes nonnegative integer values. The parameter, λ, is both the mean and the variance of the distribution. Thus, as the size of the numbers in a particular sample of Poisson random numbers gets larger, so does the variability of the numbers.

The Poisson distribution is the limiting case of a binomial distribution where N approaches infinity and p goes to zero while Np = λ.

The Poisson and exponential distributions are related. If the number of counts follows the Poisson distribution, then the interval between individual counts follows the exponential distribution.

### Parameters

The MLE and the MVUE of the Poisson parameter, λ, is the sample mean. The sum of independent Poisson random variables is also Poisson distributed with the parameter equal to the sum of the individual parameters. This is used to calculate confidence intervals λ. As λ increase, the Poisson distribution can be approximated by a normal distribution with µ = λ and σ2 = λ. This approximation is used to calculate confidence intervals for values of λ greater than 100.

### Examples

#### Compute and Plot Poisson Distribution PDF

Compute and plot the pdf of a Poisson distribution with parameter `lambda = 5`.

```x = 0:15; y = poisspdf(x,5); plot(x,y,'+')```