Editing Skellam distribution (section)

{{Short description|Discrete probability distribution}}
{{Probability distribution|
  name       =Skellam|
  type       =mass|
  pdf_image  =[[Image:Skellam distribution.svg|325px|Examples of the probability mass function for the Skellam distribution.]]<br /><small>Examples of the probability mass function for the Skellam distribution. The horizontal axis is the index ''k''. (The function is only defined at integer values of ''k''. The connecting lines do not indicate continuity.)</small>|
  cdf_image  =|
  parameters =<math>\mu_1\ge 0,~~\mu_2\ge 0</math>|
  support    =<math>k \in \{\ldots, -2,-1,0,1,2,\ldots\}</math>|
  pdf        =<math>e^{-(\mu_1\!+\!\mu_2)}
\left(\frac{\mu_1}{\mu_2}\right)^{k/2}\!\!I_{k}(2\sqrt{\mu_1\mu_2})</math>|
  cdf        =|
  mean       =<math>\mu_1-\mu_2\,</math>|
  median     =N/A|
  mode       =|
  variance   =<math>\mu_1+\mu_2\,</math>|
  skewness   =<math>\frac{\mu_1-\mu_2}{(\mu_1+\mu_2)^{3/2}}</math>|
  kurtosis   =<math>\frac{1}{\mu_1+\mu_2}</math>|
  entropy    =|
  mgf        =<math>e^{-(\mu_1+\mu_2)+\mu_1e^t+\mu_2e^{-t}}</math>|
  char       =<math>e^{-(\mu_1+\mu_2)+\mu_1e^{it}+\mu_2e^{-it}}</math>
}}

The '''Skellam distribution''' is the [[discrete probability distribution]] of the difference <math>N_1-N_2</math> of two [[statistically independent]] [[random variable]]s <math>N_1</math> and <math>N_2,</math> each [[Poisson distribution|Poisson-distributed]] with respective [[expected value]]s <math>\mu_1</math> and <math>\mu_2</math>. It is useful in describing the statistics of the difference of two images with simple [[photon noise]], as well as describing the [[Spread betting|point spread]] distribution in sports where all scored points are equal, such as [[baseball]], [[ice hockey|hockey]] and [[soccer]].

The distribution is also applicable to a special case of the difference of dependent Poisson random variables, but just the obvious case where the two variables have a common additive random contribution which is cancelled by the differencing: see Karlis & Ntzoufras (2003) for details and an application.

The [[probability mass function]] for the Skellam distribution for a difference <math>K=N_1-N_2</math> between two independent Poisson-distributed random variables with means <math>\mu_1</math> and <math>\mu_2</math> is given by:

:<math>
  p(k;\mu_1,\mu_2) = \Pr\{K=k\} = e^{-(\mu_1+\mu_2)}
  \left({\mu_1\over\mu_2}\right)^{k/2}I_{k}(2\sqrt{\mu_1\mu_2})
</math>

where ''I<sub>k</sub>''(''z'') is the [[Bessel function#Modified Bessel functions : I.CE.B1.2C K.CE.B1|modified Bessel function]] of the first kind. Since ''k'' is an integer we have that ''I<sub>k</sub>''(''z'')=''I<sub>|k|</sub>''(''z'').