Investment Studio > Expressions > Functions > Statistical > POISSON

float poisson(integer events, float lambda, boolean cumulative = TRUE)

Returns the Poisson probability function.

events > 0 is the actual number of events.

lambda > 0 is the expected number of events.

If cumulative = TRUE, the CDF (Cumulative Distribution Function) is returned (equal to the probability that events is >= a stochastic variable with Poisson distribution); otherwise, the PDF (Probability Density Function) is returned. If omitted, cumulative defaults to TRUE.

The Poisson PDF is

f(events, lambda) = (exp(-events) * lambda^events) / fact(events)

It can be derived as the continuous limit of the binomial distribution for large numbers of trials with small success probability (see binomdist). The CDF is

Stochastic processes satisfying the following conditions are known as Poisson processes:

1. The numbers of events in non-overlapping intervals are independent for all intervals.
2. The probability of an event in a sufficiently small interval is proportional to the interval length.
3. The probability of two or more events in a sufficiently small interval is essentially zero.

The Poisson distribution describes the probability of observing events in a given interval when the number of trials becomes large.

Note that the parameter lambda has the same interpretation as in the exponential distribution: if the distribution is over time, it's the event intensity (= events / time unit).

Example

If a piece of radioactive material undergoes (on average) l = 120 decays an hour, the probability of at most 1 decay (i.e. 0 or 1 decays) occurring in any given 1-second interval is

=poisson(1, 120 / (60 * 60))

» 99.95%. The probability of exactly one decay occurring is

=poisson(1, 120 / (60 * 60), FALSE)

» 3.2%.

See also expondist, gammadist.