Editing Gamma distribution (section)

{{Short description|Probability distribution}}
{{Infobox probability distribution 2
| name       = Gamma
| type       = density
| pdf_image  = [[Image:Gammapdf252.svg|325px|Probability density plots of gamma distributions]]
| cdf_image  = [[Image:Gammacdf252.svg|325px|Cumulative distribution plots of gamma distributions]]
| parameters =
* {{math|''α'' > 0}} [[shape parameter|shape]]
* {{math|''θ'' > 0}} [[scale parameter|scale]]
| support    = <math>x \in [0, \infty)</math>
| pdf        = <math>f(x)=\frac{1}{\Gamma(\alpha) \theta^\alpha} x^{\alpha - 1} e^{-x/\theta}</math>
| cdf        = <math>F(x)=\frac{1}{\Gamma(\alpha)} \gamma\left(\alpha, \frac{x}{\theta}\right)</math>
| mean       = <math>\alpha \theta </math>
| median     = No simple closed form
| mode       = <math>(\alpha - 1)\theta \text{ for } \alpha \geq 1</math>, <math>0 \text{ for } \alpha < 1</math>
| variance   = <math>\alpha \theta^2</math>
| skewness   = <math>\frac{2}{\sqrt{\alpha}}</math>
| kurtosis   = <math>\frac{6}{\alpha}</math>
| entropy    = <math>\begin{align}
                      \alpha &+ \ln\theta + \ln\Gamma(\alpha)\\
                        &+ (1 - \alpha)\psi(\alpha)
                    \end{align}</math>
| mgf        = <math>(1 - \theta t)^{-\alpha} \text{ for } t < \frac{1}{\theta}</math>
| char       = <math>(1 - \theta it)^{-\alpha}</math>
| parameters2 = {{bulleted list
 | {{math|''α'' > 0}} [[shape parameter|shape]]
 | {{math|''λ'' > 0}} [[rate parameter|rate]]
 }}
| support2    = <math>x \in (0, \infty)</math>
| pdf2        = <math>f(x)=\frac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\lambda x }</math>
| cdf2        = <math>F(x)=\frac{1}{\Gamma(\alpha)} \gamma(\alpha, \lambda x)</math>
| mean2       = <math>\frac{\alpha}{\lambda}</math>
| median2     = No simple closed form
| mode2       = <math>\frac{\alpha - 1}{\lambda} \text{ for } \alpha \geq 1\text{, }0 \text{ for } \alpha < 1</math>
| variance2   = <math>\frac{\alpha}{\lambda^2}</math>
| skewness2   = <math>\frac{2}{\sqrt{\alpha}}</math>
| kurtosis2   = <math>\frac{6}{\alpha}</math>
| entropy2    = <math>\begin{align}
                      \alpha &- \ln \lambda + \ln\Gamma(\alpha)\\
                             &+ (1 - \alpha)\psi(\alpha)
                    \end{align}</math>
| mgf2        = <math>\left(1 - \frac{t}{\lambda}\right)^{-\alpha} \text{ for } t < \lambda</math>
| char2       = <math>\left(1 - \frac{it}{\lambda}\right)^{-\alpha}</math>
| moments     = <math> \alpha = \frac{E[X]^2}{V[X]}, </math> <math> \theta = \frac{V[X]}{E[X]} \quad \quad</math>
| moments2    = <math> \alpha = \frac{E[X]^2}{V[X]}, </math> <math>\lambda = \frac{E[X]}{V[X]} </math>
| fisher      = <math>I(\alpha, \theta) = \begin{pmatrix}\psi^{(1)}(\alpha) & \theta^{-1} \\ \theta^{-1} & \alpha \theta^{-2}\end{pmatrix}</math>
| fisher2     = <math>I(\alpha, \lambda) = \begin{pmatrix}\psi^{(1)}(\alpha) & -\lambda^{-1} \\ -\lambda^{-1} & \alpha \lambda^{-2}\end{pmatrix}</math>
}}

In [[probability theory]] and [[statistics]], the '''gamma distribution''' is a versatile two-[[statistical parameter|parameter]] family of continuous [[probability distribution]]s.<ref>{{Cite web |title=Gamma distribution {{!}} Probability, Statistics, Distribution {{!}} Britannica |url=https://www.britannica.com/science/gamma-distribution |access-date=2024-10-09 |website=www.britannica.com |language=en |archive-date=2024-05-19 |archive-url=https://web.archive.org/web/20240519084458/https://www.britannica.com/science/gamma-distribution |url-status=live }}</ref>  The [[exponential distribution]], [[Erlang distribution]], and [[chi-squared distribution]] are special cases of the gamma distribution.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Gamma Distribution |url=https://mathworld.wolfram.com/GammaDistribution.html |access-date=2024-10-09 |website=mathworld.wolfram.com |language=en |archive-date=2024-05-28 |archive-url=https://web.archive.org/web/20240528053806/https://mathworld.wolfram.com/GammaDistribution.html |url-status=live }}</ref>  There are two equivalent parameterizations in common use:
# With a [[shape parameter]] {{mvar|α}} and a [[scale parameter]] {{mvar|θ}}
# With a shape parameter <math>\alpha</math> and a [[rate parameter]] {{tmath|1=\lambda = 1/ \theta}}
In each of these forms, both parameters are positive real numbers.

The distribution has important applications in various fields, including [[econometrics]], [[Bayesian statistics]], and life testing.<ref>{{Cite web |title=Gamma Distribution {{!}} Gamma Function {{!}} Properties {{!}} PDF |url=https://www.probabilitycourse.com/chapter4/4_2_4_Gamma_distribution.php |access-date=2024-10-09 |website=www.probabilitycourse.com |archive-date=2024-06-13 |archive-url=https://web.archive.org/web/20240613044322/https://www.probabilitycourse.com/chapter4/4_2_4_Gamma_distribution.php |url-status=live }}</ref> In econometrics, the (''α'', ''θ'') parameterization is common for modeling waiting times, such as the time until death, where it often takes the form of an [[Erlang distribution]] for integer ''α'' values. Bayesian statisticians prefer the (''α'',''λ'') parameterization, utilizing the gamma distribution as a [[conjugate prior]] for several inverse scale parameters, facilitating analytical tractability in posterior distribution computations. The probability density and cumulative distribution functions of the gamma distribution vary based on the chosen parameterization, both offering insights into the behavior of gamma-distributed random variables. The gamma distribution is integral to modeling a range of phenomena due to its flexible shape, which can capture various statistical distributions, including the exponential and chi-squared distributions under specific conditions. Its mathematical properties, such as mean, variance, skewness, and higher moments, provide a toolset for statistical analysis and inference. Practical applications of the distribution span several disciplines, underscoring its importance in theoretical and applied statistics.<ref>{{Cite web |date=2019-03-11 |title=4.5: Exponential and Gamma Distributions |url=https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/MATH_345__-_Probability_(Kuter)/4:_Continuous_Random_Variables/4.5:_Exponential_and_Gamma_Distributions |access-date=2024-10-10 |website=Statistics LibreTexts |language=en}}</ref>

The gamma distribution is the [[maximum entropy probability distribution]] (both with respect to a uniform base measure and a <math>1/x</math> base measure) for a random variable {{mvar|X}} for which {{math|1='''E'''[''X''] = ''αθ'' = ''α''/''λ''}} is fixed and greater than zero, and {{math|1='''E'''[ln ''X''] = ''ψ''(''α'') + ln ''θ'' = ''ψ''(''α'') − ln ''λ''}} is fixed ({{mvar|ψ}} is the [[digamma function]]).<ref>{{cite journal |last1=Park |first1=Sung Y. |last2=Bera |first2=Anil K. |year=2009 |title=Maximum entropy autoregressive conditional heteroskedasticity model |journal=Journal of Econometrics |volume=150 |issue=2 |pages=219–230 |url=http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf |access-date=2011-06-02 |doi=10.1016/j.jeconom.2008.12.014 |citeseerx=10.1.1.511.9750 |archive-url=https://web.archive.org/web/20160307144515/http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf |archive-date=2016-03-07 |url-status=dead }}</ref>