Poisson distribution

Poisson distribution

Probability function
Distribution function
Designation
Options
Carrier
Probability function
Distribution function
Expected value
Median
Fashion
Dispersion
Asymmetry coefficient
Coefficient of kurtosis
Informational entropy
Generating function of moments
Characteristic function

The Poisson distribution is a probability distribution of a discrete type, models a random variable representing the number of events that occurred in a fixed time, provided that these events occur with a certain fixed average intensity and independently of each other.

Poisson distribution plays a key role in queuing theory.

Definition [edit]

Choose a fixed number and define a discrete distribution defined by the following probability function:

,

Where

• denotes factorial numbers ,
• - the basis of natural logarithm.

The fact that the random variable has a Poisson distribution with the parameter recorded: .

Moments [edit]

The generating Poisson distribution moment function has the form:

,

from where

,

.

For factorial distribution points, the general formula is valid:

,

Where

And since the moments and factorial moments are linearly related, it is often for the Poisson distribution that the factorial moments are investigated, from which, if necessary, ordinary moments can be derived.

Poisson distribution properties [edit]

• The sum of independent Poisson random variables also has a Poisson distribution. Let be . Then

.

• Let be and . Then the conditional distribution provided that , binomially. More accurately:

.

Asymptotic desire for distribution

Quite often in probability theory it is not the Poisson distribution itself that is considered, but a sequence of distributions that are asymptotically equal to it. More formally, consider a sequence of random variables. , taking integer values, such that for any done at .

The simplest example is the case when has a binomial distribution with probability of success in each of tests.

Feedback with factorial moments [edit]

Consider a sequence of random variables that accept integer non-negative values. If a at and any fixed (Where - factor factor), then for all at done .

As an example of a non-trivial consequence of this theorem, for example, the asymptotic tendency to the distribution of the number of isolated edges (two-vertex connected components) in a random -vertex graph, where each of the edges is included in the graph with probability . ^[one]

History [edit]

The work of Poisson "Studies on the probability of sentences in criminal and civil cases" was published in 1837. ^{[2] [3]} Examples of other situations that can be modeled using this distribution: equipment breakdowns, the duration of repair work performed by a stably working employee, a printing error, the growth of a colony of bacteria in a Petri dish, defects in a long ribbon or chain, radiation counter pulses et al. ^[4]

See also [edit]