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Poisson distribution


Poisson Distribution

Probability Mass FunctionThe 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, ...
lambda 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 FunctionThe formula for the Poisson cumulative probability function isF(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 FunctionThe 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
Meanlambda
ModeFor non-integer lambda, it is the largest integer less than lambda. For integer lambda, x = lambda and x = lambda - 1 are both the mode.
Range0 to positive infinity
Standard DeviationSQRT(lambda)
Coefficient of Variation1/SQRT(lambda)
Skewness1/SQRT(lambda)
Kurtosis
Parameter EstimationThe maximum likelihood estimator of lambda isltilde = XBAR
where XBAR is the sample mean.
SoftwareMost general purpose statistical software programs support at least some of the probability functions for the Poisson distribution.

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