statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, the displaced Poisson, also known as the hyper-Poisson distribution, is a generalization of the

Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...

. The

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

mass function is :

P(X=n) = \begin
    e^\dfrac\cdot\dfrac, \quad n=0,1,2,\ldots &\text r\geq 0\\

0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...

e^\dfrac\cdot\dfrac,\quad n=s,s+1,s+2,\ldots &\text \end where

\lambda>0

and ''r'' is a new parameter; the Poisson distribution is recovered at ''r'' = 0. Here

I\left(r,\lambda\right)

is the Pearson's

incomplete gamma function In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which ...

: :

I(r,\lambda)=\sum^\infty_\frac,

where ''s'' is the integral part of ''r''. The motivation given by Staff is that the ratio of successive probabilities in the Poisson distribution (that is

P(X=n)/P(X=n-1)

) is given by

\lambda/n

for

n>0

and the displaced Poisson generalizes this ratio to

\lambda/\left(n+r\right)

References

{{stat-stub Discrete distributions