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.

统计和机器学习olbox™ offers several ways to work with the Poisson distribution.

Create a probability distribution objectPoissonDistributionby fitting a probability distribution to sample data or by specifying parameter values. Then, use object functions to evaluate the distribution, generate random numbers, and so on.
Work with the Poisson distribution interactively by using theDistribution Fitterapp. You can export an object from the app and use the object functions.
Use distribution-specific functions (poisscdf,poisspdf,poissinv,poisstat,poissfit,poissrnd) with specified distribution parameters. The distribution-specific functions can accept parameters of multiple Poisson distributions.
Use generic distribution functions (cdf,icdf,pdf,random) with a specified distribution name ('Poisson') and parameters.

Parameters

The Poisson distribution uses the following parameter.

Parameter	Description	金宝app
`lambda`(λ)	Mean	$λ \geq 0$

The parameterλis also equal to the variance of the Poisson distribution.

The sum of two Poisson random variables with parametersλ₁andλ₂is a Poisson random variable with parameterλ=λ₁+λ₂.

Probability Density Function

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

$f (x | λ) = \frac{λ^{x}}{x!} e^{- λ}; x = 0, 1, 2, \dots, \infty .$

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

For an example, seeCompute Poisson Distribution pdf.

Cumulative Distribution Function

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

$p = F (x | λ) = e^{- λ} \sum_{i = 0}^{f l o o r (x)} \frac{λ^{i}}{i!} .$

The result is the probability of at mostxoccurrences of the random event.

For an example, seeCompute Poisson Distribution cdf.

Examples

Compute Poisson Distribution pdf

Open Live Script

Compute the pdf of the Poisson distribution with parameterlambda = 4.

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

Plot the pdf with bars of width1.

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

Figure contains an axes object. The axes object contains an object of type bar.

Compute Poisson Distribution cdf

Open Live Script

Compute the cdf of the Poisson distribution with parameterlambda = 4.

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

Plot the cdf.

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

Figure contains an axes object. The axes object contains an object of type stair.

Compare Poisson and Normal Distribution pdfs

Open Live Script

Whenlambdais large, the Poisson distribution can be approximated by the normal distribution with meanlambdaand variancelambda.

Compute the pdf of the Poisson distribution with parameterlambda = 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) holdonplot(x2,y2,'LineWidth',2) xlabel('Observation') ylabel('Probability') title('Poisson and Normal pdfs') legend('Poisson Distribution','Normal Distribution','location','northwest') holdoff

Figure contains an axes object. The axes object with title Poisson and Normal pdfs contains 2 objects of type bar, line. These objects represent Poisson Distribution, Normal Distribution.

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 inNindependent trials with the probability of successp. The Poisson distribution is the limiting case of a binomial distribution whereNapproaches infinity andpgoes to zero whileNp=λ. SeeCompare Binomial and Poisson Distribution pdfs.
Exponential Distribution— The exponential distribution is a one-parameter continuous distribution that has parameterμ(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 $\frac{1}{λ}$ .
Normal Distribution— The normal distribution is a two-parameter continuous distribution that has parametersμ(mean) andσ(standard deviation). Whenλis large, the Poisson distribution can be approximated by the normal distribution withμ=λandσ²=λ. SeeCompare 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.