## Poisson distribution

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## Distribution, Poisson

# Distribution, Poisson

The Poisson distribution (named after Siméon Denis Poisson, 1781–1840) is used to describe certain events in time or in space. It is derived from the binomial distribution with the extension that time (or space) is thought to be continuous instead of discrete. In the case of discrete time (binomial distribution), let the probability that a certain event occurs during one period of length 1 be a constant *p* ; then the number of events *Z* ∊ {0, 1, 2, …) happening during a certain number *n* of such periods is binomially distributed with parameters *n* and *p*. No event influences any other event—that is, the events are mutually independent. If the length of the period is decreased to Δ*t*, and if we are still interested in the number of events happening during a time span of length *n*, then this time span consists of *n/Δt* periods within each of which the event probability is *p* Δ*t*. For *t* → 0 and *np = λ*, the formula for the binomial distribution

is transformed into the formula of the Poisson distribution

The difference between the two distributions is shown in two graphs drawn from a sample of 25,000 simulated random events:

Random variables *X* _{t} form a Poisson process if the following conditions hold: The probability that exactly one event occurs during a time span of length Δ*t* is proportional to the length of the time span; the probability that more than one event occurs during this time span is negligible; and the number of events occurring in disjoint time intervals are mutually independent. *X* _{s} is the number of events that occurred from *t* = 0 until *t* = s. The increments (*X _{t} – X_{s}* ) follow the Poisson distribution.

The Poisson distribution is also called a “distribution of rare events,” because for a fixed λ and a large *n* the probability of the individual event occurring per time unit is, of course, small. An important application of the

Poisson distribution and of the Poisson process (a stochastic process whose random increments are Poisson distributed) is queuing theory; in this case, the number of new customers arriving during a fixed period is the random increment. The parameter *λ* of the Poisson distribution is called “intensity” and yields the expected value of customers arriving per time unit in this example.

Another example is the number of accidents that occur in a certain area during a given period. The distribution of this number will also follow the Poisson distribution. If one compares the number of car accidents with passengers killed with and without a seat belt, one has to compare the parameters of the two Poisson distributions and would find out whether using seat belts had a significant influence on the number of deadly accidents.

Given the conditions under which events are Poisson distributed, one could argue that customers arriving at an airport often arrive in pairs or even larger groups—which would violate the applicability of the model to real-world scenarios. Another obvious violation is due to the fact that the arrival intensities at airports (or in emergency units) are not the same over the entire day or the week, such that the model can be applied only to selected times spans during which the arrival intensities are more or less constant over a considerable period of time. Physical processes such as the decay of radioactive material are less prone to violations of this kind than processes where humans are involved; but even here—at least with the help of computer simulation—intensities that vary over time can be successfully modelled.

**SEE ALSO** *Central Tendencies, Measures of; Distribution, Normal; Distribution, Uniform; Variables, Random*

## BIBLIOGRAPHY

Hoel, Paul G. 1984. *Introduction to Mathematical Statistics*. 5th ed. Hoboken, NJ: Wiley.

Poisson, Siméon D. [1837] 2003. *Recherches sur la probabilité des jugements en matière criminelle et en matière civile*. Paris: Editions Jacques Gabay.

*Klaus G. Troitzsch*

## Poisson distribution

**Poisson distribution** The basic discrete probability distribution for data in the form of counts of random events. If each event occurs with the same probability and the mean frequency of events is μ, the probability that exactly *r* events will occur is e^{–μ}μ* ^{r}*/

*r*!

The Poisson distribution is discrete, taking the values

*r*= 0, 1, 2,… and it can be obtained as a limiting case of the binomial distribution as

*n*tends to infinity while

*np*is held fixed. The mean and variance of the Poisson distribution are both equal to μ.

## Poisson distribution

**Poisson distribution** The basis of a method whereby the distribution of a particular attribute in a population can be calculated from its mean occurrence in a random sample of the population, provided that the population is large and the probability that the attribute will occur is less than 0.1. For a given mean, the distribution can be calculated, giving the probability that a sample will contain 0, 1, 2, 3,… examples of the particular attribute. The distribution is named after the French mathematician S. D. Poisson (1781–1840).

## Poisson distribution

**Poisson distribution** The basis of a method whereby the distribution of a particular attribute in a population can be calculated from its mean occurrence in a random sample of that population, provided the population is large and there is a less than 0.1 probability that the attribute will occur. For a given mean the distribution can be calculated, giving the probabilities that a sample will contain 0, 1, 2, 3,…examples of the particular attribute. The distribution is named after the French mathematician S. D. Poisson (1781–1840).

## Poisson distribution

**Poisson distribution** The basis of a method whereby the distribution of a particular attribute in a population can be calculated from its mean occurrence in a random sample of the population, provided that the population is large and the probability that the attribute will occur is less than 0.1. For a given mean the distribution can be calculated, giving the probabilities that a sample will contain 0,1, 2, 3, … examples of the particular attribute. The distribution is named after the French mathematician S. D. Poisson (1781–1840).

## Poisson distribution

**Poisson distribution** A probability distribution applied in studies of rare events, such as earthquakes, or when the probability of an event taking place is small. It approximates to the binomial distribution, in certain cases, but is otherwise highly skewed.

## Poisson distribution

**Poisson distribution** In statistics, a discrete probability distribution which is applied to the number of times an event occurs.