# Poisson Distribution. Probability density function, cumulative distribution function, mean and variance

This calculator calculates poisson distribution pdf, cdf, mean and variance for given parameters

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#### Timur

Created: 2018-02-09 08:16:17, Last updated: 2021-03-06 09:47:15

In probability theory and statistics, the Poisson distribution, named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area, or volume. Wikipedia

Probability density function of the poisson distribution is
$f(k)=\frac{e^{-\lambda}\lambda^x}{x!}$,
where lambda is a parameter $0 < \lambda < \infinity$ which equals the average number of events per interval. It is also called the rate parameter.

For instance, on a particular river, overflow floods occur once every 100 years on average. If we assume the Poisson model is appropriate, we can calculate the probability of k = 0, 1, ... overflow floods in a 100-year interval using a Poisson distribution with lambda equals 1.

Cumulative distribution function of the poisson distribution is
$F(k)=e^{-\lambda }\sum _{i=0}^{\lfloor k\rfloor }{\frac {\lambda ^{i}}{i!}}\$,
where $\lfloor k\rfloor$ is the floor function

Mean or expected value for the poisson distribution is
$\mu_x=\lambda$

Variance is
$\sigma^{2}_{x}=\lambda$

The calculator below calculates the mean and variance of Poisson distribution and plots probability density function and cumulative distribution function for given parameters lambda and n - number of points to plot on the chart.

#### Poisson Distribution. Probability density function, cumulative distribution function, mean and variance

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Mean

Variance

Probability density function
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Cumulative distribution function
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PLANETCALC, Poisson Distribution. Probability density function, cumulative distribution function, mean and variance