See soorce fer Poisson distreebution

{{Probability distribution
  | name       = Poisson
  | type       = mass
  | pdf_image  = [[File:poisson pmf.svg|325px|Plot o the Poisson PMF]]<br />The horizontal axis is the index ''k'', the nummer o occurrences. ''λ'' is the expectit value. The function is defined anly at integer values o ''k''. The connectin lines are anly guides for the ee.
  | cdf_image  = [[File:poisson cdf.svg|325px|Plot o the Poisson CDF]]<br />The horizontal axis is the index ''k'', the number o occurrences. The CDF is discontinuous at the integers o ''k'' an flat everywhaur ense acause a variable that is Poisson distreebutit taks on anly integer values.
  | parameters = ''λ'' > 0 ([[real nummer|real]])
  | support    = ''k'' ∈ &#x2124;<sup>*</sup>
  | pdf        = <math>\frac{\lambda^k e^{-\lambda}}{k!}</math>
  | cdf        = <math>\frac{\Gamma(\lfloor k+1\rfloor, \lambda)}{\lfloor k\rfloor !}</math>, or <math>e^{-\lambda} \sum_{i=0}^{\lfloor k\rfloor} \frac{\lambda^i}{i!}\ </math>, or <math>Q(\lfloor k+1\rfloor,\lambda)</math>
(for <math>k\ge 0</math>, whaur <math>\Gamma(x, y)</math> is the [[incomplete gamma function]],  <math>\lfloor k\rfloor</math> is the [[fluir function]], an Q is the [[Incomplete gamma function#Regularized Gamma functions an Poisson random variables|regularised gamma function]])
  | mean       = <math>\lambda</math>
  | median     = <math>\approx\lfloor\lambda+1/3-0.02/\lambda\rfloor</math>
  | mode       = <math>\lceil\lambda\rceil - 1, \lfloor\lambda\rfloor</math>
  | variance   = <math>\lambda</math>
  | skewness   = <math>\lambda^{-1/2}</math>
  | kurtosis   = <math>\lambda^{-1}</math>
  | entropy    = <math>\lambda[1 - \log(\lambda)] + e^{-\lambda}\sum_{k=0}^\infty \frac{\lambda^k\log(k!)}{k!}</math>
(for large <math>\lambda</math>)
<math>\frac{1}{2}\log(2 \pi e \lambda) - \frac{1}{12 \lambda} - \frac{1}{24 \lambda^2} -</math><br><math>\qquad \frac{19}{360 \lambda^3} + O\left(\frac{1}{\lambda^4}\right)</math><!--formula split with \qquad indent-->
  | pgf        = <math>\exp(\lambda(z - 1))</math>
  | mgf        = <math>\exp(\lambda (e^{t} - 1))</math>
  | char       = <math>\exp(\lambda (e^{it} - 1))</math>
  | fisher     = <math>\lambda^{-1}</math>
}}
In [[probability theory]] an [[stateestics]], the '''Poisson distreebution''' (French pronunciation {{IPA-fr|pwasɔ̃|}}), named efter [[France|French]] mathematician [[Siméon Denis Poisson]], is a [[discrete probability distreebution]] that expresses the probability o a gien nummer o events occurrin in a fixed interval o time an/or space if thir events occur wi a kent average rate an [[Stateestical unthirldom|independently]] o the time syne the last event.<ref name=haight>{{cite book|author=Frank A. Haight|title=Handbook of the Poisson Distribution|publisher=John Wiley & Sons|location=New York|year=1967|ref=harv}}</ref>

==References==
{{Reflist}}

{{Authority control}}

[[Category:Discrete distreebutions]]
[[Category:Distreebutions wi conjugate priors]]
[[Category:Factorial an binomial topics]]
[[Category:Poisson processes]]
[[Category:Exponential faimily distreebutions]]
[[Category:Infinitely divisible probability distreebutions]]
[[Category:Probability distreebutions]]