Editing Erlang distribution (section)

== Characterization ==

=== Probability density function ===

The [[probability density function]] of the Erlang distribution is

:<math>f(x; k,\lambda)={\lambda^k x^{k-1} e^{-\lambda x} \over (k-1)!}\quad\mbox{for }x, \lambda \geq 0,</math>

The parameter ''k'' is called the shape parameter, and the parameter <math>\lambda</math> is called the rate parameter.

An alternative, but equivalent, parametrization uses the scale parameter <math>\beta</math>, which is the reciprocal of the rate parameter (i.e., <math>\beta = 1/\lambda</math>):

:<math>f(x; k,\beta)=\frac{ x^{k-1} e^{-\frac{x}{\beta}} }{\beta^k (k-1)!}\quad\mbox{for }x, \beta \geq 0.</math>

When the scale parameter <math>\beta</math> equals 2, the distribution simplifies to the [[chi-squared distribution]] with 2''k'' degrees of freedom. It can therefore be regarded as a [[generalized chi-squared distribution]] for even numbers of degrees of freedom.

=== Cumulative distribution function (CDF) ===

The [[cumulative distribution function]] of the Erlang distribution is

:<math>F(x; k,\lambda) = P(k, \lambda x) = \frac{\gamma(k, \lambda x)}{\Gamma(k)} = \frac{\gamma(k, \lambda x)}{(k-1)!},</math>

where <math>\gamma</math> is the lower [[incomplete gamma function]] and <math>P</math> is the [[Incomplete gamma function#Regularized Gamma functions and Poisson random variables|lower regularized gamma function]].
The CDF may also be expressed as
:<math>F(x; k,\lambda) = 1 - \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^n.</math>

=== Erlang-''k'' ===
The Erlang-''k'' distribution (where ''k'' is a positive integer) <math>E_k(\lambda)</math> is defined by setting ''k'' in the PDF of the Erlang distribution.<ref>{{Cite web |title=h1.pdf |url=https://www.win.tue.nl/~iadan/sdp/h1.pdf}}</ref> For instance, the Erlang-2 distribution is <math>E_2(\lambda) ={\lambda^2 x} e^{-\lambda x} \quad\mbox{for }x, \lambda \geq 0</math>, which is the same as <math>f(x; 2,\lambda)</math>.

===Median===
An asymptotic expansion is known for the median of an Erlang distribution,<ref>{{Cite journal | last1 = Choi | first1 = K. P. | doi = 10.1090/S0002-9939-1994-1195477-8 | title = On the medians of gamma distributions and an equation of Ramanujan | journal = Proceedings of the American Mathematical Society | volume = 121 | pages = 245–251 | year = 1994 | issue = 1 | jstor = 2160389| doi-access =  }}</ref> for which coefficients can be computed and bounds are known.<ref>{{Cite journal | last1 = Adell | first1 = J. A. | last2 = Jodrá | first2 = P. | doi = 10.1090/S0002-9947-07-04411-X | title = On a Ramanujan equation connected with the median of the gamma distribution | journal = Transactions of the American Mathematical Society | volume = 360 | issue = 7 | pages = 3631 | year = 2010


 | doi-access = free }}</ref><ref>{{Cite journal | last1 = Jodrá | first1 = P. | title = Computing the Asymptotic Expansion of the Median of the Erlang Distribution | doi = 10.3846/13926292.2012.664571 | journal = Mathematical Modelling and Analysis | volume = 17 | issue = 2 | pages = 281–292 | year = 2012 | doi-access = free }}</ref> An approximation is <math>\frac{k}{\lambda}\left(1-\dfrac{1}{3k+0.2}\right),</math> i.e. below the mean <math>\frac{k}{\lambda}.</math><ref name=Banneheka2009>{{cite journal | last1 = Banneheka | first1 = BMSG | last2 = Ekanayake | first2 = GEMUPD | year = 2009 | title = A new point estimator for the median of gamma distribution | journal = Viyodaya J Science | volume = 14 | pages = 95–103 }}</ref>