## Poisson Distribution

### Overview

The Poisson distribution is a one-parameter family of curves that models the number of times a random event occurs. This distribution is appropriate for applications that involve counting the number of times a random event occurs in a given amount of time, distance, area, and so on. 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 packets lost over a network per minute.

Statistics and Machine Learning Toolbox™ offers several ways to work with the Poisson distribution.

### Parameters

The Poisson distribution uses the following parameter.

ParameterDescriptionSupport
`lambda` (λ)Mean$\lambda \ge 0$

The parameter λ is also equal to the variance of the Poisson distribution.

The sum of two Poisson random variables with parameters λ1 and λ2 is a Poisson random variable with parameter λ = λ1 + λ2 .

### Probability Density Function

The probability density function (pdf) of the Poisson distribution is

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

The result is the probability of exactly x occurrences of the random event. For discrete distributions, the pdf is also known as the probability mass function (pdf).

For an example, see Compute Poisson Distribution pdf.

### Cumulative Distribution Function

The cumulative distribution function (cdf) of the Poisson distribution is

`$p=F\left(x|\lambda \right)={e}^{-\lambda }\sum _{i=0}^{floor\left(x\right)}\frac{{\lambda }^{i}}{i!}.$`

The result is the probability of at most x occurrences of the random event.

For an example, see Compute Poisson Distribution cdf.

### Examples

#### Compute Poisson Distribution pdf

Compute the pdf of the Poisson distribution with parameter `lambda = 4`.

```x = 0:15; y = poisspdf(x,4);```

Plot the pdf with bars of width `1`.

```figure bar(x,y,1) xlabel('Observation') ylabel('Probability')```

#### Compute Poisson Distribution cdf

Compute the cdf of the Poisson distribution with parameter `lambda = 4`.

```x = 0:15; y = poisscdf(x,4);```

Plot the cdf.

```figure stairs(x,y) xlabel('Observation') ylabel('Cumulative Probability')```

#### Compare Poisson and Normal Distribution pdfs

When `lambda` is large, the Poisson distribution can be approximated by the normal distribution with mean `lambda` and variance `lambda`.

Compute the pdf of the Poisson distribution with parameter `lambda = 50`.

```lambda = 50; x1 = 0:100; y1 = poisspdf(x1,lambda);```

Compute the pdf of the corresponding normal distribution.

```mu = lambda; sigma = sqrt(lambda); x2 = 0:0.1:100; y2 = normpdf(x2,mu,sigma);```

Plot the pdfs on the same axis.

```figure bar(x1,y1,1) hold on plot(x2,y2,'LineWidth',2) xlabel('Observation') ylabel('Probability') title('Poisson and Normal pdfs') legend('Poisson Distribution','Normal Distribution','location','northwest') hold off```

The pdf of the normal distribution closely approximates the pdf of the Poisson distribution.

### Related Distributions

• Binomial Distribution — The binomial distribution is a two-parameter discrete distribution that counts the number of successes in N independent trials with the probability of success p. The Poisson distribution is the limiting case of a binomial distribution where N approaches infinity and p goes to zero while Np = λ. See Compare Binomial and Poisson Distribution pdfs.

• Exponential Distribution — The exponential distribution is a one-parameter continuous distribution that has parameter μ (mean). The Poisson distribution models counts of the number of times a random event occurs in a given amount of time. In such a model, the amount of time between occurrences is modeled by the exponential distribution with mean $\frac{1}{\lambda }$.

• Normal Distribution — The normal distribution is a two-parameter continuous distribution that has parameters μ (mean) and σ (standard deviation). When λ is large, the Poisson distribution can be approximated by the normal distribution with μ = λ and σ2 = λ. See Compare Poisson and Normal Distribution pdfs.

## References

[1] Abramowitz, Milton, and Irene A. Stegun, eds. Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. 9. Dover print.; [Nachdr. der Ausg. von 1972]. Dover Books on Mathematics. New York, NY: Dover Publ, 2013.

[2] Devroye, Luc. Non-Uniform Random Variate Generation. New York, NY: Springer New York, 1986. https://doi.org/10.1007/978-1-4613-8643-8

[3] Evans, Merran, Nicholas Hastings, and Brian Peacock. Statistical Distributions. 2nd ed. New York: J. Wiley, 1993.

[4] Loader, Catherine. Fast and Accurate Computation of Binomial Probabilities. July 9, 2000.