Editing Gamma distribution (section)

== Definitions ==
The parameterization with {{mvar|α}} and {{mvar|θ}} appears to be more common in [[econometrics]] and other applied fields, where the gamma distribution is frequently used to model waiting times. For instance, in [[Accelerated life testing|life testing]], the waiting time until death is a [[random variable]] that is frequently modeled with a gamma distribution. See Hogg and Craig<ref>{{cite book |author-link=Robert V. Hogg |first1=R. V. |last1=Hogg |first2=A. T. |last2=Craig |year=1978 |title=Introduction to Mathematical Statistics |edition=4th |location=New York |publisher=Macmillan |isbn=0023557109|pages=Remark 3.3.1}}</ref>  for an explicit motivation.

The parameterization with {{mvar|α}} and {{mvar|λ}} is more common in [[Bayesian statistics]], where the gamma distribution is used as a [[conjugate prior]] distribution for various types of inverse scale (rate) parameters, such as the {{mvar|λ}} of an [[exponential distribution]] or a [[Poisson distribution]]<ref>{{Cite arXiv |eprint=1311.1704 |class=cs.IR |first1=Prem |last1=Gopalan |first2=Jake M. |last2=Hofman |title=Scalable Recommendation with Poisson Factorization |last3=Blei |first3=David M. |year=2013 |author3-link=David Blei}}</ref> – or for that matter, the {{mvar|λ}} of the gamma distribution itself. The closely related [[inverse-gamma distribution]] is used as a conjugate prior for scale parameters, such as the [[variance]] of a [[normal distribution]].

If {{mvar|α}} is a positive [[integer]], then the distribution represents an [[Erlang distribution]]; i.e., the sum of {{mvar|α}} independent [[exponentially distributed]] [[random variable]]s, each of which has a mean of {{mvar|θ}}.

=== Characterization using shape ''α'' and rate ''λ'' ===
The gamma distribution can be parameterized in terms of a [[shape parameter]] {{math|1=''α''}} and an inverse scale parameter {{math|1=''λ'' = 1/''θ''}}, called a [[rate parameter]]. A random variable {{mvar|X}} that is gamma-distributed with shape {{mvar|α}} and rate {{mvar|λ}} is denoted

<math display=block>X \sim \Gamma(\alpha, \lambda) \equiv \operatorname{Gamma}(\alpha,\lambda)</math>

The corresponding probability density function in the shape-rate parameterization is

<math display=block>
\begin{align}
f(x;\alpha,\lambda) & = \frac{ x^{\alpha-1} e^{-\lambda x} \lambda^\alpha}{\Gamma(\alpha)} \quad \text{ for } x > 0 \quad \alpha, \lambda > 0, \\[6pt]
\end{align}
</math>

where <math>\Gamma(\alpha)</math> is the [[gamma function]].
For all positive integers, <math>\Gamma(\alpha)=(\alpha-1)!</math>.

The [[cumulative distribution function]] is the regularized gamma function:

<math display=block> F(x;\alpha,\lambda) = \int_0^x f(u;\alpha,\lambda)\,du= \frac{\gamma(\alpha, \lambda x)}{\Gamma(\alpha)},</math>

where <math>\gamma(\alpha, \lambda x)</math> is the lower [[incomplete gamma function]].

If {{mvar|α}} is a positive [[integer]] (i.e., the distribution is an [[Erlang distribution]]), the cumulative distribution function has the following series expansion:<ref name="Papoulis"/>

<math display=block>F(x;\alpha,\lambda) = 1-\sum_{i=0}^{\alpha-1} \frac{(\lambda x)^i}{i!} e^{-\lambda x} = e^{-\lambda x} \sum_{i=\alpha}^\infty \frac{(\lambda x)^i}{i!}.</math>

=== Characterization using shape ''α'' and scale ''θ'' ===
A random variable {{mvar|X}} that is gamma-distributed with shape {{mvar|α}} and scale {{mvar|θ}} is denoted by

<math display=block>X \sim \Gamma(\alpha, \theta) \equiv \operatorname{Gamma}(\alpha, \theta)</math>

[[Image:Gamma-PDF-3D.png|thumb|right|320px|Illustration of the gamma PDF for parameter values over {{mvar|α}} and {{mvar|x}} with {{mvar|θ}} set to {{math|1, 2, 3, 4, 5,}} and&nbsp;{{math|6}}. One can see each {{mvar|θ}} layer by itself here [http://commons.wikimedia.org/wiki/File:Gamma-PDF-3D-by-k.png] as well as by&nbsp;{{mvar|α}} [http://commons.wikimedia.org/wiki/File:Gamma-PDF-3D-by-Theta.png] and&nbsp;{{mvar|x}}. [http://commons.wikimedia.org/wiki/File:Gamma-PDF-3D-by-x.png].]]

The [[probability density function]] using the shape-scale parametrization is

<math display=block>f(x;\alpha,\theta) =  \frac{x^{\alpha-1}e^{-x/\theta}}{\theta^\alpha\Gamma(\alpha)} \quad \text{ for } x > 0 \text{ and } \alpha, \theta > 0.</math>

Here {{math|Γ(''α'')}} is the [[gamma function]] evaluated at {{mvar|α}}.

The [[cumulative distribution function]] is the regularized gamma function:

<math display=block> F(x;\alpha,\theta) = \int_0^x f(u;\alpha,\theta)\,du = \frac{\gamma\left(\alpha, \frac{x}{\theta}\right)}{\Gamma(\alpha)},</math>

where <math>\gamma\left(\alpha, \frac{x}{\theta}\right)</math> is the lower [[incomplete gamma function]].

It can also be expressed as follows, if {{mvar|α}} is a positive [[integer]] (i.e., the distribution is an [[Erlang distribution]]):<ref name="Papoulis">Papoulis, Pillai, ''Probability, Random Variables, and Stochastic Processes'', Fourth Edition</ref>

<math display=block>F(x;\alpha,\theta) = 1-\sum_{i=0}^{\alpha-1} \frac{1}{i!} \left(\frac{x}{\theta} \right)^i e^{-x/\theta} = e^{-x/\theta} \sum_{i=\alpha}^\infty \frac{1}{i!} \left( \frac{x}{\theta} \right)^i.</math>

Both parametrizations are common because either can be more convenient depending on the situation.