Poisson distribution

Poisson Distribution

Probability Mass Function

The Poisson distribution is used to model the number of events occurring within a given time interval.The formula for the Poisson probability mass function is
p(x,lambda) = EXP(-lambda)*lambda**(x)/x! for x = 0, 1, 2, ...

p(x,lambda) = EXP(-lambda)*lambda**(x)/x! for x = 0, 1, 2, ...

is the shape parameter which indicates the average number of events in the given time interval.
The following is the plot of the Poisson probability density function for four values of lambda

plot of the Poisson probability density function

Cumulative Distribution Function

The formula for the Poisson cumulative probability function is F(x,lambda) = SUM[EXP(-lambda)*lambda**i/i!]
where the summation is for i = 0 to x

The following is the plot of the Poisson cumulative distribution function with the same values of lambda

as the pdf plots above.
plot of the Poisson cumulative distribution function

Percent Point Function

The Poisson percent point function does not exist in simple closed form. It is computed numerically. Note that because this is a discrete distribution that is only defined for integer values of x, the percent point function is not smooth in the way the percent point function typically is for a continuous distribution.The following is the plot of the Poisson percent point function with the same values of lambda

as the pdf plots above.
plot of the Poisson percent point function

Common Statistics

Mean
Mode	For non-integer , it is the largest integer less than . For integer , x = and x = - 1 are both the mode.
Range	0 to positive infinity
Standard Deviation
Coefficient of Variation
Skewness
Kurtosis

Parameter Estimation

The maximum likelihood estimator of lambda

where

is the sample mean.

Software

Most general purpose statistical software programs support at least some of the probability functions for the Poisson distribution.

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