Poisson distribution

Pois·son distribution

P0404500 (pwä-sôN′)n. Statistics A probability distribution which arises when counting the number of occurrences of a rare event in a long series of trials.

[After Siméon Denis Poisson (1781-1840), French mathematician.]

Poisson distribution

(ˈpwɑːsən) n (Statistics) statistics a distribution that represents the number of events occurring randomly in a fixed time at an average rate λ; symbol P0(λ). For large n and small p with np = λ it approximates to the binomial distribution Bi(n,p)[C19: named after S. D. Poisson]

Pois•son′ distribu`tion

(pwɑˈsoʊn, -ˈsɔ̃)
n. a probability distribution whose mean and variance are identical. [1920–25; after S. Dutch. Poisson (1781–1840), French mathematician and physicist]

Thesaurus

Noun

Poisson distribution - a theoretical distribution that is a good approximation to the binomial distribution when the probability is small and the number of trials is largedistribution, statistical distribution - (statistics) an arrangement of values of a variable showing their observed or theoretical frequency of occurrencestatistics - a branch of applied mathematics concerned with the collection and interpretation of quantitative data and the use of probability theory to estimate population parameters

Poisson distribution

[pwä′sōn‚dis·trə′byü·shən] (statistics) A probability distribution whose mean and variance have a common value k, and whose frequency is ƒ(x) = k ^x e ^-k/ x !, for x = 0,1,2,….

Poisson Distribution

one of the most important probability distributions of random variables that assume integral values. A random variable X that obeys a Poisson distribution takes on only nonnegative values; the probability that X = k is

where λ is a positive parameter. The distribution is named after S. D. Poisson; it first appeared in a work by him published in 1837. The mathematical expectation and variance of a random variable having a Poisson distribution with the parameter λ are equal to λ. If the independent random variables X₁ and X₂ have Poisson distributions with parameters λ₁ and λ₂, their sum X₁ + X₂ has a Poisson distribution with the parameter λ₁ + λ₂. Illustrations of Poisson distributions are given in Figure 1.

Figure 1

In probability-theoretic models, the Poisson distribution is used both as an approximating and an exact distribution. For example, if the events A₁,…, A_n occur in n independent trials with the same low probability p, the probability that k of the n events simultaneously occur is approximated by the function p_k(np); the mathematical significance of this assertion for large n and 1/p is stated by Poisson’s theorem. In particular, such a model is an excellent description of the process of radioactive decay and of numerous other physical phenomena.

The Poisson distribution is used as an exact distribution in the theory of stochastic processes. An example is the calculation of the load on communication lines. The number of calls made in nonoverlapping intervals of time are usually assumed to be independent random variables that obey a Poisson distribution with parameters whose values are proportional to the lengths of the corresponding intervals.

The arithmetic mean X̄ = (X₁ + … + X_n)/n of n observed values of the random variables X₁,…, X_n is used as an estimate of the unknown parameter λ, since this estimate is unbiased and has a minimum standard deviation.

REFERENCES

Gnedenko, B. V. Kurs teorii veroiatnostei, 5th ed. Moscow-Leningrad, 1969.
Feller, W. Vvedenie v teoriiu veroiatnostei i ee prilozheniia, 2nd ed., vol. 1. Moscow, 1967. (Translated from English.)

Poisson distribution

(mathematics)A probability distribution used to describethe occurrence of unlikely events in a large number ofindependent trials.

Poisson distributions are often used in building simulateduser loads.

Poisson distribution

A statistical method developed by the 18th century French mathematician S. D. Poisson, which is used for predicting the probable distribution of a series of events. For example, when the average transaction volume in a communications system can be estimated, Poisson distribution is used to determine the probable minimum and maximum number of transactions that can occur within a given time period.

Poisson distribution

Pois·son dis·tri·bu·tion

(pwah-son[h]'), 1. a discontinuous distribution important in statistical work and defined by the equation p (x) = e ^-μμ^x/ x!, where e is the base of natural logarithms, x is the sequence of integers, μ is the mean, and x! represents the factorial of x. 2. a distribution function used to describe the occurrence of rare events, or the sampling distribution of isolated counts in a continuum of time or space.

Poisson distribution

A sampling distribution based on the number of occurrences, r, of an event during a period of time, which depends on only one parameter, the mean number of occurrences in periods of the same length.

Poisson distribution

Statistics The distribution that arises when parasites are distributed randomly among hosts. See Distribution.

Poisson distribution

(statistics) the frequency of sample classes containing a particular number of events (0,1,2,3 … n), where the average frequency of the event is small in relation to the total number of times that the event could occur. Thus, if a pool contained 100 small fish then each time a net is dipped into the pool up to 100 fish could be caught and returned to the pool. In reality, however, only none, one or two fish are likely to be caught each time. The Poisson distribution predicts the probability of catching 0,1,2,3 … 100 fish each time, producing a FREQUENCY DISTRIBUTION graph that is skewed heavily towards the low number of events.

Poisson,

Siméon Denis, French mathematician, 1781-1840. Poisson distribution - a discontinuous distribution important in statistical work.Poisson ratioPoisson-Pearson formula - to determines the statistical error in calculating the endemic index of malaria.

Poisson distribution

Related to Poisson distribution: binomial distribution, Poisson process

noun

Words related to Poisson distribution

noun a theoretical distribution that is a good approximation to the binomial distribution when the probability is small and the number of trials is large

Related Words

distribution
statistical distribution
statistics