Editing Gamma distribution (section)

=== 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.