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».

Poisson(10)→

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

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.