Editing Erlang distribution (section)

{{short description|Family of continuous probability distributions}}
{{about|the mathematical / statistical distribution concept||Erlang (disambiguation){{!}}Erlang}}
{{Multiple issues|
{{more citations needed|date=June 2012}}
{{more footnotes|date=June 2012}}
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{{Probability distribution
|name       =Erlang
|type       =density
|pdf_image  =[[File:Erlang dist pdf2.svg|325px|Probability density plots of Erlang distributions]]
|cdf_image  =[[File:Erlang dist cdf2.svg|325px|Cumulative distribution plots of Erlang distributions]]
|parameters =<math> k \in \{1,2,3,\ldots\},</math> [[shape parameter|shape]] <br /><math> \lambda \in (0,\infty),</math> rate <br />alt.: <math> \beta = 1/\lambda,</math> [[scale parameter|scale]]
|support    =<math> x \in [0, \infty)</math>
|pdf        =<math> \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!}</math>
|cdf        =<math> P(k, \lambda x) = \frac{\gamma(k, \lambda x)}{(k - 1)!} = 1 - \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^{n}</math>
|mean       =<math> \frac{k}{\lambda}</math>
|mode       =<math> \frac{1}{\lambda}(k - 1)</math>
|variance   =<math> \frac{k}{\lambda^2}</math>
|median     =No simple closed form
|skewness   =<math> \frac{2}{\sqrt{k}}</math>
<!-- invalid parameter |scv        =<math> \frac{1}{k}</math> -->
|kurtosis   =<math> \frac{6}{k}</math>
|entropy    =<math> (1 - k)\psi(k) + \ln\left[\frac{\Gamma(k)}{\lambda}\right] + k</math>
|mgf        =<math> \left(1 - \frac{t}{\lambda}\right)^{-k}</math> for <math> t < \lambda</math>
|char       =<math> \left(1 - \frac{it}{\lambda}\right)^{-k}</math>|
}}
The '''Erlang distribution''' is a two-parameter family of continuous [[probability distribution]]s with [[Support (mathematics)|support]] <math> x \in [0, \infty)</math>. The two parameters are:
* a positive integer <math>k,</math> the "shape", and
* a positive real number <math>\lambda,</math> the "rate". The "scale", <math>\beta,</math> the reciprocal of the rate, is sometimes used instead.

The Erlang distribution is the distribution of a sum of <math>k</math> [[Independence (probability theory)|independent]] [[exponential distribution|exponential variables]] with mean <math>1/\lambda</math> each.  Equivalently, it is the distribution of the time until the ''k''th event of a [[Poisson process]] with a rate of <math>\lambda</math>.  The Erlang and Poisson distributions are complementary, in that while the Poisson distribution counts the events that occur in a fixed amount of time, the Erlang distribution counts the amount of time until the occurrence of a fixed number of events.  When <math>k=1</math>, the distribution  simplifies to the [[exponential distribution]]. The Erlang distribution is a special case of the [[gamma distribution]] in which the shape of the distribution is discretized.

The Erlang distribution was developed by [[Agner Krarup Erlang|A. K. Erlang]] to examine the number of telephone calls that might be made at the same time to the operators of the switching stations. This work on telephone [[Teletraffic engineering|traffic engineering]] has been expanded to consider waiting times in [[queueing theory|queueing system]]s in general. The distribution is also used in the field of [[stochastic process]]es.