Poisson distribution

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Poisson(mean)

The Poisson distribution represents the number of independent discrete random events that occur in a fixed period of time. The parameter mean specifies the expected number of events in one time unit.

You might use the Poisson distribution to model the number of sales per month of a low-volume product, or the number of airplane crashes per year. Poisson is occassionally applied to non-time intervals, such as the number of cancerous cells in a given (small) volume of tissue, or the number of mutations in a given length of DNA.

The poisson distribution assumes that each event occurs randomly and independently of all other events. When the number of events follows a Poisson(mean:m) distribution, then the time between individual events follows an Exponential(rate:1/m) distribution.

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Distributions

Details

The variance of a Poisson(mean:m) distribution is m, the standard deviation is sqrt(m), the skewness is 1/sqrt(m) and the kurtosis is 1/m. The probability density is given by

P(k|m) = Exp(-m) * mean^k / Factorial(k)

The cumulative probability for integer k>0 is:

F(k|m) = GammaFn( k+1,mean ) / Factorial(k)


For large mean values, the Poisson(m) distribution approaches a Round(Normal(m,Sqrt(m)))distribution. This approximation is extremely close for m>50 and pretty close for m>10. Of course, the Normal is a continuous distribution and the Poisson a discrete, hence a Round function is shown.

Parameter Estimation

Suppose you have historical data, Data, indexed by I, in the form of a positive count at each measurement, and you wish to estimate the parameter for the best-fit Poisson distribution. The parameter can be estimated using:

Mean(Data,I)

See Also

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