# Poisson distribution

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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.

`Poisson(15)`

→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. The Poisson distribution is occasionally 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.

## Functions

### Poisson(mean)

The distribution function. Describes an uncertain quantity that follows a Poisson distribution with a mean of «mean», or an arrival rate of 1/«mean».

### ProbPoisson(k, mean)

The probability of the outcome «k». This is the discrete probability function for Poisson. Returns a value equal to (with [math]\displaystyle{ m }[/math] denoting «mean»)

- [math]\displaystyle{ p(k) = {{m^k e^{-m}}\over{k!}} }[/math]

### CumPoisson(k, mean)

The cumulative distribution function for Poisson. Returns the probability that the outcome is less than or equal to «k».

- [math]\displaystyle{ F(k) = \sum_{k=0}^{k} {{m^i e^{-m}}\over{i!}} = {{GammaI( k+1, m )}\over{k!}} }[/math]

`CumPoisson(13,10) → 0.864`

### CumPoissonInv(p, mean)

The inverse cumulative probability function, or *quantile function*, for the Poisson distribution. Returns the smallest number of events, [math]\displaystyle{ k }[/math], such that the probability [math]\displaystyle{ P( n \le k) \ge p }[/math] when [math]\displaystyle{ n }[/math] is distributed as a Poisson(«mean»).

`CumPoissonInv(0.8, 10) → 13`

## 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)`

These are computed by the Prob_Poisson and CumPoisson functions.

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)`

## Positive Poisson distribution

The Positive Poisson Distribution, also known as the * Zero-Truncated Poisson (ZTP) distribution* and the

*, is the distribution obtained by conditioning on the constraint that the value is not zero. Hence, the sample consists of the positive integers.*

**conditional Poisson distribution**To generate a sample from the Positive Poisson distribution, using the pre-truncated mean «m», use the following expression

`Poisson(m + Ln(1 - Uniform(0,1)*(1 - Exp(-m)))) + 1`

## History

The analytic distribution functions, ProbPoisson, CumPoisson and CumPoissonInv, were added as built-in functions in Analytica 5.2. In releases prior to 5.2, you have to add the Distribution Densities Library to your model.

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