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.

Comments

Double exponential distribution

Double Exponential Distribution Probability Density Function The general formula for the probability density function of the double exponential distribution is where is the location parameter and is the scale parameter . The case where = 0 and = 1 is called the standard double exponential distribution . The equation for the standard double exponential distribution is Since the general form of probability functions can be expressed in terms of the standard distribution , all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the double exponential probability density function. Cumulative Distribution Function The formula for the cumulative distribution function of the double exponential distribution is The following is the plot of the double exponential cumulative distribution function. Percent Point Function The formula for the percent point function of the double exponential distribution

Runs Test for Detecting Non-randomness

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Basics of Sampling Techniques

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