Editing Gamma distribution (section)

=== 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>