gamma distribution

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 life testing, the waiting time until death is a random variable that is frequently modeled with a gamma distribution. See Hogg and Craig{{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}} 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{{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}} – 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 variables, each of which has a mean of {{mvar|θ}}.

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

$$X\; \backslash sim\; \backslash Gamma(\backslash alpha,\; \backslash lambda)\; \backslash equiv\; \backslash operatorname\{Gamma\}(\backslash alpha,\backslash lambda)$$

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

\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}

where $\backslash Gamma(\backslash alpha)$ is the gamma function.

For all positive integers, $\backslash Gamma(\backslash alpha)=(\backslash alpha-1)!$.

The cumulative distribution function is the regularized gamma function:

$$F(x;\backslash alpha,\backslash lambda)\; =\; \backslash int\_0^x\; f(u;\backslash alpha,\backslash lambda)\backslash ,du=\; \backslash frac\{\backslash gamma(\backslash alpha,\; \backslash lambda\; x)\}\{\backslash Gamma(\backslash alpha)\},$$

where $\backslash gamma(\backslash alpha,\; \backslash lambda\; x)$ 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:

$$F(x;\backslash alpha,\backslash lambda)\; =\; 1-\backslash sum\_\{i=0\}^\{\backslash alpha-1\}\; \backslash frac\{(\backslash lambda\; x)^i\}\{i!\}\; e^\{-\backslash lambda\; x\}\; =\; e^\{-\backslash lambda\; x\}\; \backslash sum\_\{i=\backslash alpha\}^\backslash infty\; \backslash frac\{(\backslash lambda\; x)^i\}\{i!\}.$$

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

$$X\; \backslash sim\; \backslash Gamma(\backslash alpha,\; \backslash theta)\; \backslash equiv\; \backslash operatorname\{Gamma\}(\backslash alpha,\; \backslash theta)$$

Image:Gamma-PDF-3D.png

The probability density function using the shape-scale parametrization is

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

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

The cumulative distribution function is the regularized gamma function:

$$F(x;\backslash alpha,\backslash theta)\; =\; \backslash int\_0^x\; f(u;\backslash alpha,\backslash theta)\backslash ,du\; =\; \backslash frac\{\backslash gamma\backslash left(\backslash alpha,\; \backslash frac\{x\}\{\backslash theta\}\backslash right)\}\{\backslash Gamma(\backslash alpha)\},$$

where $\backslash gamma\backslash left(\backslash alpha,\; \backslash frac\{x\}\{\backslash theta\}\backslash right)$ 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):Papoulis, Pillai, Probability, Random Variables, and Stochastic Processes, Fourth Edition

$$F(x;\backslash alpha,\backslash theta)\; =\; 1-\backslash sum\_\{i=0\}^\{\backslash alpha-1\}\; \backslash frac\{1\}\{i!\}\; \backslash left(\backslash frac\{x\}\{\backslash theta\}\; \backslash right)^i\; e^\{-x/\backslash theta\}\; =\; e^\{-x/\backslash theta\}\; \backslash sum\_\{i=\backslash alpha\}^\backslash infty\; \backslash frac\{1\}\{i!\}\; \backslash left(\; \backslash frac\{x\}\{\backslash theta\}\; \backslash right)^i.$$

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

The mean of gamma distribution is given by the product of its shape and scale parameters:

$$\backslash mu\; =\; \backslash alpha\backslash theta\; =\; \backslash alpha/\backslash lambda$$

The variance is:

$$\backslash sigma^2\; =\; \backslash alpha\; \backslash theta^2\; =\; \backslash alpha/\backslash lambda^2$$

The square root of the inverse shape parameter gives the coefficient of variation:

$$\backslash sigma/\backslash mu\; =\; \backslash alpha^\{-0.5\}\; =\; 1/\backslash sqrt\{\backslash alpha\}$$

The skewness of the gamma distribution only depends on its shape parameter, {{mvar|α}}, and it is equal to $2/\backslash sqrt\{\backslash alpha\}.$

The {{mvar|n}}-th raw moment is given by:

\mathrm{E}[X^n] = \theta^n \frac{\Gamma(\alpha+n)}{\Gamma(\alpha)} = \theta^n \prod_{i=1}^n(\alpha+i-1) \; \text{ for } n=1, 2, \ldots.

File:Gamma distribution median bounds.png

Unlike the mode and the mean, which have readily calculable formulas based on the parameters, the median does not have a closed-form equation. The median for this distribution is the value $\backslash nu$ such that

$$\backslash frac\{1\}\{\backslash Gamma(\backslash alpha)\; \backslash theta^\backslash alpha\}\; \backslash int\_0^\{\backslash nu\}\; x^\{\backslash alpha\; -\; 1\}\; e^\{-x/\backslash theta\}\; dx\; =\; \backslash frac\{1\}\{2\}.$$

A rigorous treatment of the problem of determining an asymptotic expansion and bounds for the median of the gamma distribution was handled first by Chen and Rubin, who proved that (for $\backslash theta\; =\; 1$)

where $\backslash mu(\backslash alpha)\; =\; \backslash alpha$ is the mean and $\backslash nu(\backslash alpha)$ is the median of the $\backslash text\{Gamma\}(\backslash alpha,1)$ distribution.Jeesen Chen, Herman Rubin, Bounds for the difference between median and mean of gamma and Poisson distributions, Statistics & Probability Letters, Volume 4, Issue 6, October 1986, Pages 281–283, {{issn|0167-7152}}, [https://dx.doi.org/10.1016/0167-7152(86)90044-1] {{Webarchive|url=https://web.archive.org/web/20241009203229/https://www.sciencedirect.com/unsupported_browser|date=2024-10-09}}. For other values of the scale parameter, the mean scales to $\backslash mu\; =\; \backslash alpha\backslash theta$, and the median bounds and approximations would be similarly scaled by {{mvar|θ}}.

K. P. Choi found the first five terms in a Laurent series asymptotic approximation of the median by comparing the median to Ramanujan's $\backslash theta$ function.Choi, K. P. [https://www.ams.org/journals/proc/1994-121-01/S0002-9939-1994-1195477-8/S0002-9939-1994-1195477-8.pdf "On the Medians of the Gamma Distributions and an Equation of Ramanujan"] {{Webarchive|url=https://web.archive.org/web/20210123121523/https://www.ams.org/journals/proc/1994-121-01/S0002-9939-1994-1195477-8/S0002-9939-1994-1195477-8.pdf |date=2021-01-23 }}, Proceedings of the American Mathematical Society, Vol. 121, No. 1 (May, 1994), pp. 245–251. Berg and Pedersen found more terms:{{cite journal |author=Berg, Christian |author2=Pedersen, Henrik L. |name-list-style=amp |title=The Chen–Rubin conjecture in a continuous setting |journal=Methods and Applications of Analysis |date=March 2006 |volume=13 |issue=1 |pages=63–88 |doi=10.4310/MAA.2006.v13.n1.a4 |s2cid=6704865 |url=https://www.intlpress.com/site/pub/files/_fulltext/journals/maa/2006/0013/0001/MAA-2006-0013-0001-a004.pdf |access-date=1 April 2020 |doi-access=free |archive-date=16 January 2021 |archive-url=https://web.archive.org/web/20210116114105/https://www.intlpress.com/site/pub/files/_fulltext/journals/maa/2006/0013/0001/MAA-2006-0013-0001-a004.pdf |url-status=live }}

$$\backslash nu(\backslash alpha)\; =\; \backslash alpha\; -\; \backslash frac\{1\}\{3\}\; +\; \backslash frac\{8\}\{405\backslash alpha\}\; +\; \backslash frac\{184\}\{25515\; \backslash alpha^2\}\; +\; \backslash frac\{2248\}\{3444525\; \backslash alpha^3\}\; -\; \backslash frac\{19006408\}\{15345358875\; \backslash alpha^4\}\; -\; O\backslash left(\backslash frac\{1\}\{\backslash alpha^5\}\backslash right)\; +\; \backslash cdots$$

File:Gamma distribution median Lyon bounds.png

File:Gamma distribution median loglog bounds.png of upper (solid) and lower (dashed) bounds to the median of a gamma distribution and the gaps between them. The green, yellow, and cyan regions represent the gap before the Lyon 2021 paper. The green and yellow narrow that gap with the lower bounds that Lyon proved. Lyon's bounds proved in 2023 further narrow the yellow. Mostly within the yellow, closed-form rational-function-interpolated conjectured bounds are plotted along with the numerically calculated median (dotted) value. Tighter interpolated bounds exist but are not plotted, as they would not be resolved at this scale.]]

Partial sums of these series are good approximations for high enough {{mvar|α}}; they are not plotted in the figure, which is focused on the low-{{mvar|α}} region that is less well approximated.

Berg and Pedersen also proved many properties of the median, showing that it is a convex function of {{mvar|α}},Berg, Christian and Pedersen, Henrik L. [https://arxiv.org/abs/math/0609442 "Convexity of the median in the gamma distribution"] {{Webarchive|url=https://web.archive.org/web/20230526181721/https://arxiv.org/abs/math/0609442 |date=2023-05-26 }}. and that the asymptotic behavior near $\backslash alpha\; =\; 0$ is $\backslash nu(\backslash alpha)\; \backslash approx\; e^\{-\backslash gamma\}2^\{-1/\backslash alpha\}$ (where {{mvar|γ}} is the Euler–Mascheroni constant), and that for all $\backslash alpha\; >\; 0$ the median is bounded by .

A closer linear upper bound, for $\backslash alpha\; \backslash ge\; 1$ only, was provided in 2021 by Gaunt and Merkle,{{cite journal |last1=Gaunt, Robert E., and Milan Merkle |title=On bounds for the mode and median of the generalized hyperbolic and related distributions |journal=Journal of Mathematical Analysis and Applications |date=2021 |volume=493 |issue=1 |pages=124508|doi=10.1016/j.jmaa.2020.124508 |arxiv=2002.01884 |s2cid=221103640 }} relying on the Berg and Pedersen result that the slope of $\backslash nu(\backslash alpha)$ is everywhere less than 1:

$$\backslash nu(\backslash alpha)\; \backslash le\; \backslash alpha\; -\; 1\; +\; \backslash log2\; ~~$$ for $\backslash alpha\; \backslash ge\; 1$ (with equality at $\backslash alpha\; =\; 1$)

which can be extended to a bound for all $\backslash alpha\; >\; 0$ by taking the max with the chord shown in the figure, since the median was proved convex.

An approximation to the median that is asymptotically accurate at high {{mvar|α}} and reasonable down to $\backslash alpha\; =\; 0.5$ or a bit lower follows from the Wilson–Hilferty transformation:

$$\backslash nu(\backslash alpha)\; =\; \backslash alpha\; \backslash left(\; 1\; -\; \backslash frac\{1\}\{9\backslash alpha\}\; \backslash right)^3$$

which goes negative for .

In 2021, Lyon proposed several approximations of the form $\backslash nu(\backslash alpha)\; \backslash approx\; 2^\{-1/\backslash alpha\}(A\; +\; B\backslash alpha)$. He conjectured values of {{mvar|A}} and {{mvar|B}} for which this approximation is an asymptotically tight upper or lower bound for all $\backslash alpha\; >\; 0$.{{cite journal |last1=Lyon |first1=Richard F. |title=On closed-form tight bounds and approximations for the median of a gamma distribution |journal=PLOS One |date=13 May 2021 |volume=16 |issue=5 |pages=e0251626 |doi=10.1371/journal.pone.0251626 |pmid=33984053 |pmc=8118309 |arxiv=2011.04060 |bibcode=2021PLoSO..1651626L |doi-access=free }} In particular, he proposed these closed-form bounds, which he proved in 2023:{{cite journal |last1=Lyon |first1=Richard F. |title=Tight bounds for the median of a gamma distribution |journal=PLOS One |date=13 May 2021 |volume=18 |issue=9 |pages=e0288601 |doi=10.1371/journal.pone.0288601 |pmid=37682854 |pmc=10490949 |doi-access=free }}

$$\backslash nu\_\{L\backslash infty\}(\backslash alpha)\; =\; 2^\{-1/\backslash alpha\}(\backslash log\; 2\; -\; \backslash frac\{1\}\{3\}\; +\; \backslash alpha)\; \backslash quad$$ is a lower bound, asymptotically tight as $\backslash alpha\; \backslash to\; \backslash infty$

$$\backslash nu\_U(\backslash alpha)\; =\; 2^\{-1/\backslash alpha\}(e^\{-\backslash gamma\}\; +\; \backslash alpha)\; \backslash quad$$ is an upper bound, asymptotically tight as $\backslash alpha\; \backslash to\; 0$

Lyon also showed (informally in 2021, rigorously in 2023) two other lower bounds that are not closed-form expressions, including this one involving the gamma function, based on solving the integral expression substituting 1 for $e^\{-x\}$:

$$\backslash nu(\backslash alpha)\; >\; \backslash left(\; \backslash frac\{2\}\{\backslash Gamma(\backslash alpha+1)\}\; \backslash right)^\{-1/\backslash alpha\}\; \backslash quad$$ (approaching equality as $k\; \backslash to\; 0$)

and the tangent line at $\backslash alpha\; =\; 1$ where the derivative was found to be $\backslash nu^\backslash prime(1)\; \backslash approx\; 0.9680448$:

$$\backslash nu(\backslash alpha)\; \backslash ge\; \backslash nu(1)\; +\; (\backslash alpha-1)\; \backslash nu^\backslash prime(1)\; \backslash quad$$ (with equality at $k\; =\; 1$)

$$\backslash nu(\backslash alpha)\; \backslash ge\; \backslash log\; 2\; +\; (\backslash alpha-1)\; (\backslash gamma\; -\; 2\; \backslash operatorname\{Ei\}(-\backslash log\; 2)\; -\; \backslash log\; \backslash log\; 2)$$

where Ei is the exponential integral.

Additionally, he showed that interpolations between bounds could provide excellent approximations or tighter bounds to the median, including an approximation that is exact at $\backslash alpha\; =\; 1$ (where $\backslash nu(1)\; =\; \backslash log\; 2$) and has a maximum relative error less than 0.6%. Interpolated approximations and bounds are all of the form

$$\backslash nu(\backslash alpha)\; \backslash approx\; \backslash tilde\{g\}(\backslash alpha)\backslash nu\_\{L\backslash infty\}(\backslash alpha)\; +\; (1\; -\; \backslash tilde\{g\}(\backslash alpha))\; \backslash nu\_U(\backslash alpha)$$

where $\backslash tilde\{g\}$ is an interpolating function running monotonially from 0 at low {{mvar|α}} to 1 at high {{mvar|α}}, approximating an ideal, or exact, interpolator $g(\backslash alpha)$:

$$g(\backslash alpha)\; =\; \backslash frac\{\backslash nu\_U(\backslash alpha)\; -\; \backslash nu(\backslash alpha)\}\{\backslash nu\_U(\backslash alpha)\; -\; \backslash nu\_\{L\backslash infty\}(\backslash alpha)\}$$

For the simplest interpolating function considered, a first-order rational function

$$\backslash tilde\{g\}\_1(\backslash alpha)\; =\; \backslash frac\{\backslash alpha\}\{b\_0\; +\; \backslash alpha\}$$

the tightest lower bound has

$$b\_0\; =\; \backslash frac\{\backslash frac\{8\}\{405\}\; +\; e^\{-\backslash gamma\}\; \backslash log\; 2\; -\; \backslash frac\{\backslash log^2\; 2\}\{2\}\}\{e^\{-\backslash gamma\}\; -\; \backslash log\; 2\; +\; \backslash frac\{1\}\{3\}\}\; -\; \backslash log\; 2\; \backslash approx\; 0.143472$$

and the tightest upper bound has

$$b\_0\; =\; \backslash frac\{e^\{-\backslash gamma\}\; -\; \backslash log\; 2\; +\; \backslash frac\{1\}\{3\}\}\{1\; -\; \backslash frac\{e^\{-\backslash gamma\}\; \backslash pi^2\}\{12\}\}\; \backslash approx\; 0.374654$$

The interpolated bounds are plotted (mostly inside the yellow region) in the log–log plot shown. Even tighter bounds are available using different interpolating functions, but not usually with closed-form parameters like these.

If {{math|X_i}} has a {{math|Gamma(α_i, θ)}} distribution for {{math|1=i = 1, 2, ..., N}} (i.e., all distributions have the same scale parameter {{mvar|θ}}), then

$$\backslash sum\_\{i=1\}^N\; X\_i\; \backslash sim\backslash mathrm\{Gamma\}\; \backslash left(\; \backslash sum\_\{i=1\}^N\; \backslash alpha\_i,\; \backslash theta\; \backslash right)$$

provided all {{math|X_i}} are independent.

For the cases where the {{math|X_i}} are independent but have different scale parameters, see Mathai {{Cite journal|last=Mathai|first=A. M.|title=Storage capacity of a dam with gamma type inputs|journal=Annals of the Institute of Statistical Mathematics|language=en|volume=34|issue=3|pages=591–597|doi=10.1007/BF02481056|issn=0020-3157|year=1982|s2cid=122537756}} or Moschopoulos.{{cite journal |first=P. G. |last=Moschopoulos |year=1985 |title=The distribution of the sum of independent gamma random variables |journal=Annals of the Institute of Statistical Mathematics |volume=37 |issue=3 |pages=541–544 |doi=10.1007/BF02481123 |s2cid=120066454 }}

The gamma distribution exhibits infinite divisibility.

If

$$X\; \backslash sim\; \backslash mathrm\{Gamma\}(\backslash alpha,\; \backslash theta),$$

then, for any {{math|c > 0}},

$$cX\; \backslash sim\; \backslash mathrm\{Gamma\}(\backslash alpha,\; c\backslash ,\backslash theta),$$ by moment generating functions,

or equivalently, if

$$X\; \backslash sim\; \backslash mathrm\{Gamma\}\backslash left(\; \backslash alpha,\backslash lambda\; \backslash right)$$ (shape-rate parameterization)

$$cX\; \backslash sim\; \backslash mathrm\{Gamma\}\backslash left(\; \backslash alpha,\; \backslash frac\; \backslash lambda\; c\; \backslash right),$$

Indeed, we know that if {{mvar|X}} is an exponential r.v. with rate {{mvar|λ}}, then {{math|cX}} is an exponential r.v. with rate {{math|λ/c}}; the same thing is valid with Gamma variates (and this can be checked using the moment-generating function, see, e.g.,[http://www.stat.washington.edu/thompson/S341_10/Notes/week4.pdf these notes], 10.4-(ii)): multiplication by a positive constant {{mvar|c}} divides the rate (or, equivalently, multiplies the scale).

The gamma distribution is a two-parameter exponential family with natural parameters {{math|α − 1}} and {{math|−1/θ}} (equivalently, {{math|α − 1}} and {{math|−λ}}), and natural statistics {{mvar|X}} and {{math|ln X}}.

If the shape parameter {{mvar|α}} is held fixed, the resulting one-parameter family of distributions is a natural exponential family.

One can show that

$$\backslash operatorname\{E\}[\backslash ln\; X]\; =\; \backslash psi(\backslash alpha)\; -\; \backslash ln\; \backslash lambda$$

or equivalently,

$$\backslash operatorname\{E\}[\backslash ln\; X]\; =\; \backslash psi(\backslash alpha)\; +\; \backslash ln\; \backslash theta$$

where {{mvar|ψ}} is the digamma function. Likewise,

$$\backslash operatorname\{var\}[\backslash ln\; X]\; =\; \backslash psi^\{(1)\}(\backslash alpha)$$

where $\backslash psi^\{(1)\}$ is the trigamma function.

This can be derived using the exponential family formula for the moment generating function of the sufficient statistic, because one of the sufficient statistics of the gamma distribution is {{math|ln x}}.

The information entropy is

\begin{align}

\operatorname{H}(X) & = \operatorname{E}[-\ln p(X)] \\[4pt]

& = \operatorname{E}[-\alpha \ln \lambda + \ln \Gamma(\alpha) - (\alpha-1)\ln X + \lambda X] \\[4pt]

& = \alpha - \ln \lambda + \ln \Gamma(\alpha) + (1-\alpha)\psi(\alpha).

\end{align}

In the {{mvar|α}}, {{mvar|θ}} parameterization, the information entropy is given by

$$\backslash operatorname\{H\}(X)\; =\backslash alpha\; +\; \backslash ln\; \backslash theta\; +\; \backslash ln\; \backslash Gamma(\backslash alpha)\; +\; (1-\backslash alpha)\backslash psi(\backslash alpha).$$

Image:Gamma-KL-3D.png

The Kullback–Leibler divergence (KL-divergence), of {{math|Gamma(α_p, λ_p)}} ("true" distribution) from {{math|Gamma(α_q, λ_q)}} ("approximating" distribution) is given by{{cite web|first=W. D.|last=Penny|url=https://www.fil.ion.ucl.ac.uk/~wpenny/publications/densities.ps|title= KL-Divergences of Normal, Gamma, Dirichlet, and Wishart densities}}

\begin{align}

D_{\mathrm{KL}}(\alpha_p,\lambda_p; \alpha_q, \lambda_q) = {} & (\alpha_p-\alpha_q) \psi(\alpha_p) - \log\Gamma(\alpha_p) + \log\Gamma(\alpha_q) \\

& {} + \alpha_q(\log \lambda_p - \log \lambda_q) + \alpha_p\frac{\lambda_q-\lambda_p}{\lambda_p}.

\end{align}

Written using the {{mvar|α}}, {{mvar|θ}} parameterization, the KL-divergence of {{math|Gamma(α_p, θ_p)}} from {{math|Gamma(α_q, θ_q)}} is given by

\begin{align}

D_{\mathrm{KL}}(\alpha_p,\theta_p; \alpha_q, \theta_q) = {} & (\alpha_p-\alpha_q)\psi(\alpha_p) - \log\Gamma(\alpha_p) + \log\Gamma(\alpha_q) \\

& {} + \alpha_q(\log \theta_q - \log \theta_p) + \alpha_p \frac{\theta_p - \theta_q}{\theta_q}.

\end{align}

The Laplace transform of the gamma PDF, which is the moment-generating function of the gamma distribution, is

$$F(s)\; =\; \backslash operatorname\; E\backslash left(\; e^\{sX\}\; \backslash right)\; =\; \backslash frac1\; \{(1\; -\; \backslash theta\; s)^\backslash alpha\}\; =\; \backslash left(\; \backslash frac\backslash lambda\{\; \backslash lambda\; -\; s\}\; \backslash right)^\backslash alpha$$

(where $X$ is a random variable with that distribution).

• Let $X\_1,\; X\_2,\; \backslash ldots,\; X\_n$ be $n$ independent and identically distributed random variables following an exponential distribution with rate parameter λ, then $\backslash sum\_i\; X\_i\; \backslash sim\; \backslash operatorname\{Gamma\}(n,\backslash lambda)$ where n is the shape parameter and {{mvar|λ}} is the rate, and $\backslash bar\{X\}\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_i\; X\_i\; \backslash sim\; \backslash operatorname\{Gamma\}(n,\; n\backslash lambda)$.
• If {{math|X ~ Gamma(1, λ)}} (in the shape–rate parametrization), then {{mvar|X}} has an exponential distribution with rate parameter {{mvar|λ}}. In the shape-scale parametrization, {{math|X ~ Gamma(1, θ)}} has an exponential distribution with rate parameter {{math|1/θ}}.
• If {{math|X ~ Gamma(ν/2, 2)}} (in the shape–scale parametrization), then {{mvar|X}} is identical to {{math|χ²(ν)}}, the chi-squared distribution with {{mvar|ν}} degrees of freedom. Conversely, if {{math|Q ~ χ²(ν)}} and {{mvar|c}} is a positive constant, then {{math|cQ ~ Gamma(ν/2, 2c)}}.
• If {{math|1=θ = 1/α}}, one obtains the Schulz-Zimm distribution, which is most prominently used to model polymer chain lengths.
• If {{mvar|α}} is an integer, the gamma distribution is an Erlang distribution and is the probability distribution of the waiting time until the {{mvar|α}}-th "arrival" in a one-dimensional Poisson process with intensity {{math|1/θ}}. If

::$X\; \backslash sim\; \backslash Gamma(\backslash alpha\; \backslash in\; \backslash mathbf\{Z\},\; \backslash theta),\; \backslash qquad\; Y\; \backslash sim\; \backslash operatorname\{Pois\}\backslash left(\backslash frac\; x\; \backslash theta\; \backslash right),$

:then

::

• If {{mvar|X}} has a Maxwell–Boltzmann distribution with parameter {{mvar|a}}, then

::$X^2\; \backslash sim\; \backslash Gamma\backslash left(\backslash frac\{3\}\{2\},\; 2a^2\backslash right).$

If the shape parameter of the gamma distribution is known, but the inverse-scale parameter is unknown, then a gamma distribution for the inverse scale forms a conjugate prior. The compound distribution, which results from integrating out the inverse scale, has a closed-form solution known as the compound gamma distribution.{{cite journal|last=Dubey|first=Satya D. | title=Compound gamma, beta and F distributions|journal=Metrika|date=December 1970|volume=16|issue=1 |pages=27–31 |doi=10.1007/BF02613934|s2cid=123366328 }}

If, instead, the shape parameter is known but the mean is unknown, with the prior of the mean being given by another gamma distribution, then it results in K-distribution.

The gamma distribution $f(x;\backslash alpha)\; \backslash ,\; (\backslash alpha\; >\; 1)$ can be expressed as the product distribution of a Weibull distribution and a variant form of the stable count distribution.

Its shape parameter $\backslash alpha$ can be regarded as the inverse of Lévy's stability parameter in the stable count distribution:

f(x;\alpha) =

\int_0^\infty \frac{1}{u} \, W_k\left(\frac{x}{u}\right)

\left[ k u^{\alpha-1} \, \mathfrak{N}_{\frac{1}{\alpha}}\left(u^\alpha\right) \right] \, du ,

where $\backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)$ is a standard stable count distribution of shape $\backslash alpha$, and $W\_\backslash alpha(x)$ is a standard Weibull distribution of shape $\backslash alpha$.

The likelihood function for {{mvar|N}} iid observations {{math|(x₁, ..., x_N)}} is

$$L(\backslash alpha,\; \backslash theta)\; =\; \backslash prod\_\{i=1\}^N\; f(x\_i;\backslash alpha,\backslash theta)$$

from which we calculate the log-likelihood function

$$\backslash ell(\backslash alpha,\; \backslash theta)\; =\; (\backslash alpha\; -\; 1)\; \backslash sum\_\{i=1\}^N\; \backslash ln\; x\_i\; -\; \backslash sum\_\{i=1\}^N\; \backslash frac\{x\_i\}\; \backslash theta\; -\; N\backslash alpha\backslash ln\; \backslash theta\; -\; N\backslash ln\; \backslash Gamma(\backslash alpha)$$

Finding the maximum with respect to {{mvar|θ}} by taking the derivative and setting it equal to zero yields the maximum likelihood estimator of the {{mvar|θ}} parameter, which equals the sample mean $\backslash bar\{x\}$ divided by the shape parameter {{mvar|α}}:

$$\backslash hat\{\backslash theta\}\; =\; \backslash frac\{1\}\{\backslash alpha\; N\}\backslash sum\_\{i=1\}^N\; x\_i\; =\; \backslash frac\{\backslash bar\{x\}\}\{\backslash alpha\}$$

Substituting this into the log-likelihood function gives

$$\backslash ell(\backslash alpha)\; =\; (\backslash alpha-1)\backslash sum\_\{i=1\}^N\; \backslash ln\; x\_i\; -N\backslash alpha\; -\; N\backslash alpha\backslash ln\; \backslash left(\backslash frac\{\backslash sum\; x\_i\}\{\backslash alpha\; N\}\; \backslash right)\; -\; N\backslash ln\; \backslash Gamma(\backslash alpha)$$

We need at least two samples: $N\backslash ge2$, because for $N=1$, the function $\backslash ell(\backslash alpha)$ increases without bounds as $\backslash alpha\backslash to\backslash infty$. For $\backslash alpha>0$, it can be verified that $\backslash ell(\backslash alpha)$ is strictly concave, by using inequality properties of the polygamma function. Finding the maximum with respect to {{mvar|α}} by taking the derivative and setting it equal to zero yields

$$\backslash ln\; \backslash alpha\; -\; \backslash psi(\backslash alpha)\; =\; \backslash ln\backslash left(\backslash frac\{1\}\{N\}\backslash sum\_\{i=1\}^N\; x\_i\backslash right)\; -\; \backslash frac\; 1\; N\; \backslash sum\_\{i=1\}^N\; \backslash ln\; x\_i\; =\; \backslash ln\; \backslash bar\{x\}\; -\; \backslash overline\{\backslash ln\; x\}$$

where {{mvar|ψ}} is the digamma function and $\backslash overline\{\backslash ln\; x\}$ is the sample mean of {{math|ln x}}. There is no closed-form solution for {{mvar|α}}. The function is numerically very well behaved, so if a numerical solution is desired, it can be found using, for example, Newton's method. An initial value of {{mvar|k}} can be found either using the method of moments, or using the approximation

$$\backslash ln\; \backslash alpha\; -\; \backslash psi(\backslash alpha)\; \backslash approx\; \backslash frac\{1\}\{2\backslash alpha\}\backslash left(1\; +\; \backslash frac\{1\}\{6\backslash alpha\; +\; 1\}\; \backslash right)$$

If we let

$$s\; =\; \backslash ln\; \backslash left(\backslash frac\; 1\; N\; \backslash sum\_\{i=1\}^N\; x\_i\backslash right)\; -\; \backslash frac\; 1\; N\; \backslash sum\_\{i=1\}^N\; \backslash ln\; x\_i\; =\; \backslash ln\; \backslash bar\{x\}\; -\; \backslash overline\{\backslash ln\; x\}$$

then {{mvar|α}} is approximately

$$k\; \backslash approx\; \backslash frac\{3\; -\; s\; +\; \backslash sqrt\{(s\; -\; 3)^2\; +\; 24s\}\}\{12s\}$$

which is within 1.5% of the correct value.{{cite web |last=Minka |first=Thomas P. |year=2002 |title=Estimating a Gamma distribution |url=https://tminka.github.io/papers/minka-gamma.pdf }} An explicit form for the Newton–Raphson update of this initial guess is:{{cite journal |last1=Choi |first1=S. C. |last2=Wette |first2=R. |year=1969 |title=Maximum Likelihood Estimation of the Parameters of the Gamma Distribution and Their Bias |journal=Technometrics |volume=11 |issue=4 |pages=683–690 |doi=10.1080/00401706.1969.10490731 }}

$$\backslash alpha\; \backslash leftarrow\; \backslash alpha\; -\; \backslash frac\{\; \backslash ln\; \backslash alpha\; -\; \backslash psi(k)\; -\; s\; \}\{\; \backslash frac\; 1\; \backslash alpha\; -\; \backslash psi\backslash prime(\backslash alpha)\; \}.$$

At the maximum-likelihood estimate $(\backslash hat\; \backslash alpha,\backslash hat\backslash theta)$, the expected values for {{mvar|x}} and $\backslash ln\; x$ agree with the empirical averages:

\begin{align}

\hat \alpha\hat\theta &= \bar x &&\text{and} &

\psi(\hat \alpha)+\ln \hat\theta &= \overline{\ln x}.

\end{align}

For data, $(x\_1,\backslash ldots,x\_N)$, that is represented in a floating point format that underflows to 0 for values smaller than $\backslash varepsilon$, the logarithms that are needed for the maximum-likelihood estimate will cause failure if there are any underflows. If we assume the data was generated by a gamma distribution with cdf $F(x;\backslash alpha,\backslash theta)$, then the probability that there is at least one underflow is:

P(\text{underflow}) = 1-(1-F(\varepsilon;\alpha,\theta))^N

This probability will approach 1 for small {{mvar|α}} and large {{mvar|N}}. For example, at $\backslash alpha=10^\{-2\}$, $N=10^4$ and $\backslash varepsilon=2.25\backslash times10^\{-308\}$, $P(\backslash text\{underflow\})\backslash approx\; 0.9998$. A workaround is to instead have the data in logarithmic format.

In order to test an implementation of a maximum-likelihood estimator that takes logarithmic data as input, it is useful to be able to generate non-underflowing logarithms of random gamma variates, when . Following the implementation in scipy.stats.loggamma, this can be done as follows: sample $Y\backslash sim\backslash text\{Gamma\}(\backslash alpha+1,\backslash theta)$ and $U\backslash sim\backslash text\{Uniform\}$ independently. Then the required logarithmic sample is $Z=\backslash ln(Y)+\backslash ln(U)/\backslash alpha$, so that $\backslash exp(Z)\backslash sim\backslash text\{Gamma\}(k,\backslash theta)$.

There exist consistent closed-form estimators of {{mvar|α}} and {{mvar|θ}} that are derived from the likelihood of the generalized gamma distribution.{{cite journal | url=https://amstat.tandfonline.com/doi/abs/10.1080/00031305.2016.1209129 | doi=10.1080/00031305.2016.1209129 | title=Closed-Form Estimators for the Gamma Distribution Derived from Likelihood Equations | year=2017 | last1=Ye | first1=Zhi-Sheng | last2=Chen | first2=Nan | journal=The American Statistician | volume=71 | issue=2 | pages=177–181 | s2cid=124682698 | access-date=2019-07-27 | archive-date=2023-05-26 | archive-url=https://web.archive.org/web/20230526181723/https://amstat.tandfonline.com/doi/abs/10.1080/00031305.2016.1209129 | url-status=live | url-access=subscription }}

The estimate for the shape {{mvar|α}} is

$$\backslash hat\{\backslash alpha\}\; =\; \backslash frac\{N\; \backslash sum\_\{i=1\}^N\; x\_i\}\{N\; \backslash sum\_\{i=1\}^N\; x\_i\; \backslash ln\; x\_i\; -\; \backslash sum\_\{i=1\}^N\; x\_i\; \backslash sum\_\{i=1\}^N\; \backslash ln\; x\_i\}$$

and the estimate for the scale {{mvar|θ}} is

$$\backslash hat\{\backslash theta\}\; =\; \backslash frac\{1\}\{N^2\}\; \backslash left(N\; \backslash sum\_\{i=1\}^N\; x\_i\; \backslash ln\; x\_i\; -\; \backslash sum\_\{i=1\}^N\; x\_i\; \backslash sum\_\{i=1\}^N\; \backslash ln\; x\_i\backslash right)$$

Using the sample mean of {{mvar|x}}, the sample mean of {{math|ln x}}, and the sample mean of the product {{math|x·ln x}} simplifies the expressions to:

$$\backslash hat\{\backslash alpha\}\; =\; \backslash bar\{x\}\; /\; \backslash hat\{\backslash theta\}$$

$$\backslash hat\{\backslash theta\}\; =\; \backslash overline\{x\backslash ln\; x\}\; -\; \backslash bar\{x\}\; \backslash overline\{\backslash ln\; x\}.$$

If the rate parameterization is used, the estimate of $\backslash hat\{\backslash lambda\}\; =\; 1/\backslash hat\{\backslash theta\}$.

These estimators are not strictly maximum likelihood estimators, but are instead referred to as mixed type log-moment estimators. They have however similar efficiency as the maximum likelihood estimators.

Although these estimators are consistent, they have a small bias. A bias-corrected variant of the estimator for the scale {{mvar|θ}} is

$$\backslash tilde\{\backslash theta\}\; =\; \backslash frac\{N\}\{N\; -\; 1\}\; \backslash hat\{\backslash theta\}$$

A bias correction for the shape parameter {{mvar|α}} is given as{{cite journal | url=https://www.tandfonline.com/doi/abs/10.1080/00031305.2018.1513376 | doi=10.1080/00031305.2018.1513376 | title=A Note on Bias of Closed-Form Estimators for the Gamma Distribution Derived from Likelihood Equations | year=2019 | last1=Louzada | first1=Francisco | last2=Ramos | first2=Pedro L. | last3=Ramos | first3=Eduardo | journal=The American Statistician | volume=73 | issue=2 | pages=195–199 | s2cid=126086375 | access-date=2019-07-27 | archive-date=2023-05-26 | archive-url=https://web.archive.org/web/20230526181723/https://www.tandfonline.com/doi/abs/10.1080/00031305.2018.1513376 | url-status=live | url-access=subscription | hdl=11449/184543 | hdl-access=free }}

$$\backslash tilde\{\backslash alpha\}\; =\; \backslash hat\{\backslash alpha\}\; -\; \backslash frac\{1\}\{N\}\; \backslash left(3\; \backslash hat\{\backslash alpha\}\; -\; \backslash frac\{2\}\{3\}\; \backslash left(\backslash frac\{\backslash hat\{\backslash alpha\}\}\{1\; +\; \backslash hat\{\backslash alpha\}\}\backslash right)\; -\; \backslash frac\{4\}\{5\}\; \backslash frac\{\backslash hat\{\backslash alpha\}\}\{(1\; +\; \backslash hat\{\backslash alpha\})^2\}\; \backslash right)$$

With known {{mvar|α}} and unknown {{mvar|θ}}, the posterior density function for theta (using the standard scale-invariant prior for {{mvar|θ}}) is

$$P(\backslash theta\; \backslash mid\; \backslash alpha,\; x\_1,\; \backslash dots,\; x\_N)\; \backslash propto\; \backslash frac\; 1\; \backslash theta\; \backslash prod\_\{i=1\}^N\; f(x\_i;\; \backslash alpha,\; \backslash theta)$$

Denoting

$$y\; \backslash equiv\; \backslash sum\_\{i=1\}^Nx\_i\; ,\; \backslash qquad\; P(\backslash theta\; \backslash mid\; \backslash alpha,\; x\_1,\; \backslash dots,\; x\_N)\; =\; C(x\_i)\; \backslash theta^\{-N\; \backslash alpha-1\}\; e^\{-y/\backslash theta\}$$

where the {{mvar|C}} (integration) constant does not depend on {{mvar|θ}}. The form of the posterior density reveals that {{math|1 / θ}} is gamma-distributed with shape parameter {{math|Nα + 2}} and rate parameter {{mvar|y}}. Integration with respect to {{mvar|θ}} can be carried out using a change of variables to find the integration constant

$$\backslash int\_0^\backslash infty\; \backslash theta^\{-N\backslash alpha\; -\; 1\; +\; m\}\; e^\{-y/\backslash theta\}\backslash ,\; d\backslash theta\; =\; \backslash int\_0^\backslash infty\; x^\{N\backslash alpha\; -\; 1\; -\; m\}\; e^\{-xy\}\; \backslash ,\; dx\; =\; y^\{-(N\backslash alpha\; -\; m)\}\; \backslash Gamma(N\backslash alpha\; -\; m)\; \backslash !$$

The moments can be computed by taking the ratio ({{mvar|m}} by {{math|1=m = 0}})

$$\backslash operatorname\{E\}\; [x^m]\; =\; \backslash frac\; \{\backslash Gamma\; (N\backslash alpha\; -\; m)\}\; \{\backslash Gamma(N\backslash alpha)\}\; y^m$$

which shows that the mean ± standard deviation estimate of the posterior distribution for {{mvar|θ}} is

$$\backslash frac\; y\; \{N\backslash alpha\; -\; 1\}\; \backslash pm\; \backslash sqrt\{\backslash frac\; \{y^2\}\; \{(N\backslash alpha\; -\; 1)^2\; (N\backslash alpha\; -\; 2)\}\}.$$

In Bayesian inference, the gamma distribution is the conjugate prior to many likelihood distributions: the Poisson, exponential, normal (with known mean), Pareto, gamma with known shape {{mvar|σ}}, inverse gamma with known shape parameter, and Gompertz with known scale parameter.

The gamma distribution's conjugate prior is:Fink, D. 1995 [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.157.5540&rep=rep1&type=pdf A Compendium of Conjugate Priors]. In progress report: Extension and enhancement of methods for setting data quality objectives. (DOE contract 95‑831).

$$p(\backslash alpha,\backslash theta\; \backslash mid\; p,\; q,\; r,\; s)\; =\; \backslash frac\{1\}\{Z\}\; \backslash frac\{p^\{\backslash alpha-1\}\; e^\{-\backslash theta^\{-1\}\; q\}\}\{\backslash Gamma(\backslash alpha)^r\; \backslash theta^\{\backslash alpha\; s\}\},$$

where {{mvar|Z}} is the normalizing constant with no closed-form solution.

The posterior distribution can be found by updating the parameters as follows:

$$\backslash begin\{align\}$$

p' &= p\prod\nolimits_i x_i,\\

q' &= q + \sum\nolimits_i x_i,\\

r' &= r + n,\\

s' &= s + n,

\end{align}

where {{mvar|n}} is the number of observations, and {{math|x_i}} is the {{mvar|i}}-th observation from the gamma distribution.

Consider a sequence of events, with the waiting time for each event being an exponential distribution with rate {{mvar|λ}}. Then the waiting time for the {{mvar|n}}-th event to occur is the gamma distribution with integer shape $\backslash alpha\; =\; n$. This construction of the gamma distribution allows it to model a wide variety of phenomena where several sub-events, each taking time with exponential distribution, must happen in sequence for a major event to occur.{{Cite book |last=Jessica. |first=Scheiner, Samuel M., 1956- Gurevitch |url=http://worldcat.org/oclc/43694448 |title=Design and analysis of ecological experiments |date=2001 |publisher=Oxford University Press |isbn=0-19-513187-8 |chapter=13. Failure-time analysis |oclc=43694448 |chapter-url=https://books.google.com/books?id=AgsTDAAAQBAJ&dq=gamma+distribution+failure+waiting+time&pg=PA235 |access-date=2022-05-26 |archive-date=2024-10-09 |archive-url=https://web.archive.org/web/20241009203230/https://search.worldcat.org/title/43694448 |url-status=live }} Examples include the waiting time of cell-division events,{{Cite journal |last=Golubev |first=A. |date=March 2016 |title=Applications and implications of the exponentially modified gamma distribution as a model for time variabilities related to cell proliferation and gene expression |url=http://dx.doi.org/10.1016/j.jtbi.2015.12.027 |journal=Journal of Theoretical Biology |volume=393 |pages=203–217 |doi=10.1016/j.jtbi.2015.12.027 |pmid=26780652 |bibcode=2016JThBi.393..203G |issn=0022-5193 |access-date=2022-05-26 |archive-date=2024-10-09 |archive-url=https://web.archive.org/web/20241009203230/https://www.sciencedirect.com/unsupported_browser |url-status=live |url-access=subscription }} number of compensatory mutations for a given mutation,{{Cite journal |last1=Poon |first1=Art |last2=Davis |first2=Bradley H |last3=Chao |first3=Lin |date=2005-07-01 |title=The Coupon Collector and the Suppressor Mutation |url=http://dx.doi.org/10.1534/genetics.104.037259 |journal=Genetics |volume=170 |issue=3 |pages=1323–1332 |doi=10.1534/genetics.104.037259 |pmid=15879511 |pmc=1451182 |issn=1943-2631 |access-date=2022-05-26 |archive-date=2024-10-09 |archive-url=https://web.archive.org/web/20241009203231/https://academic.oup.com/genetics/article/170/3/1323/6060337 |url-status=live }} waiting time until a repair is necessary for a hydraulic system,{{Cite journal |last1=Vineyard |first1=Michael |last2=Amoako-Gyampah |first2=Kwasi |last3=Meredith |first3=Jack R |date=July 1999 |title=Failure rate distributions for flexible manufacturing systems: An empirical study |url=http://dx.doi.org/10.1016/s0377-2217(98)00096-4 |journal=European Journal of Operational Research |volume=116 |issue=1 |pages=139–155 |doi=10.1016/s0377-2217(98)00096-4 |issn=0377-2217 |access-date=2022-05-26 |archive-date=2024-10-09 |archive-url=https://web.archive.org/web/20241009203230/https://www.sciencedirect.com/unsupported_browser |url-status=live |url-access=subscription }} and so on.

In biophysics, the dwell time between steps of a molecular motor like ATP synthase is nearly exponential at constant ATP concentration, revealing that each step of the motor takes a single ATP hydrolysis. If there were n ATP hydrolysis events, then it would be a gamma distribution with degree n.{{Cite journal |last1=Rief |first1=Matthias |last2=Rock |first2=Ronald S. |last3=Mehta |first3=Amit D. |last4=Mooseker |first4=Mark S. |last5=Cheney |first5=Richard E. |last6=Spudich |first6=James A. |date=2000-08-15 |title=Myosin-V stepping kinetics: A molecular model for processivity |journal=Proceedings of the National Academy of Sciences |language=en |volume=97 |issue=17 |pages=9482–9486 |doi=10.1073/pnas.97.17.9482 |issn=0027-8424 |pmc=16890 |pmid=10944217 |doi-access=free |bibcode=2000PNAS...97.9482R }}

The gamma distribution has been used to model the size of insurance claimsp. 43, Philip J. Boland, Statistical and Probabilistic Methods in Actuarial Science, Chapman & Hall CRC 2007 and rainfalls.{{Cite journal |last=Wilks |first=Daniel S. |date=1990 |title=Maximum Likelihood Estimation for the Gamma Distribution Using Data Containing Zeros |journal=Journal of Climate |volume=3 |issue=12 |pages=1495–1501 |doi=10.1175/1520-0442(1990)003<1495:MLEFTG>2.0.CO;2 |jstor=26196366 |bibcode=1990JCli....3.1495W |issn=0894-8755|doi-access=free }} This means that aggregate insurance claims and the amount of rainfall accumulated in a reservoir are modelled by a gamma process – much like the exponential distribution generates a Poisson process.

The gamma distribution is also used to model errors in multi-level Poisson regression models because a mixture of Poisson distributions with gamma-distributed rates has a known closed form distribution, called negative binomial.

In wireless communication, the gamma distribution is used to model the multi-path fading of signal power;{{citation needed|date=May 2019}} see also Rayleigh distribution and Rician distribution.

In oncology, the age distribution of cancer incidence often follows the gamma distribution, wherein the shape and scale parameters predict, respectively, the number of driver events and the time interval between them.{{cite journal |last1=Belikov |first1=Aleksey V. |title=The number of key carcinogenic events can be predicted from cancer incidence |journal=Scientific Reports |date=22 September 2017 |volume=7 |issue=1 |pages=12170 |doi=10.1038/s41598-017-12448-7|pmid=28939880 |pmc=5610194 |bibcode=2017NatSR...712170B }}{{Cite journal|last1=Belikov|first1=Aleksey V.|last2=Vyatkin|first2=Alexey|last3=Leonov|first3=Sergey V.|date=2021-08-06|title=The Erlang distribution approximates the age distribution of incidence of childhood and young adulthood cancers|journal=PeerJ|language=en|volume=9|pages=e11976|doi=10.7717/peerj.11976|pmid=34434669|pmc=8351573|issn=2167-8359|doi-access=free}}

In neuroscience, the gamma distribution is often used to describe the distribution of inter-spike intervals.J. G. Robson and J. B. Troy, "Nature of the maintained discharge of Q, X, and Y retinal ganglion cells of the cat", J. Opt. Soc. Am. A 4, 2301–2307 (1987)M.C.M. Wright, I.M. Winter, J.J. Forster, S. Bleeck "Response to best-frequency tone bursts in the ventral cochlear nucleus is governed by ordered inter-spike interval statistics", Hearing Research 317 (2014)

In bacterial gene expression where protein production can occur in bursts, the copy number of a given protein often follows the gamma distribution, where the shape and scale parameters are, respectively, the mean number of bursts per cell cycle and the mean number of protein molecules produced per burst.N. Friedman, L. Cai and X. S. Xie (2006) "Linking stochastic dynamics to population distribution: An analytical framework of gene expression", Phys. Rev. Lett. 97, 168302.

In genomics, the gamma distribution was applied in peak calling step (i.e., in recognition of signal) in ChIP-chipDJ Reiss, MT Facciotti and NS Baliga (2008) [https://web.archive.org/web/20121117144623/http://bioinformatics.oxfordjournals.org/content/24/3/396.full.pdf+html "Model-based deconvolution of genome-wide DNA binding"], Bioinformatics, 24, 396–403 and ChIP-seqMA Mendoza-Parra, M Nowicka, W Van Gool, H Gronemeyer (2013) [http://www.biomedcentral.com/1471-2164/14/834 "Characterising ChIP-seq binding patterns by model-based peak shape deconvolution"] {{Webarchive|url=https://web.archive.org/web/20241009203230/https://bmcgenomics.biomedcentral.com/articles/10.1186/1471-2164-14-834 |date=2024-10-09 }}, BMC Genomics, 14:834 data analysis.

In Bayesian statistics, the gamma distribution is widely used as a conjugate prior. It is the conjugate prior for the precision (i.e. inverse of the variance) of a normal distribution. It is also the conjugate prior for the exponential distribution.

In phylogenetics, the gamma distribution is the most commonly used approach to model among-sites rate variation{{Cite journal |last=Yang |first=Ziheng |date=September 1996 |title=Among-site rate variation and its impact on phylogenetic analyses |url=https://linkinghub.elsevier.com/retrieve/pii/0169534796100410 |journal=Trends in Ecology & Evolution |language=en |volume=11 |issue=9 |pages=367–372 |doi=10.1016/0169-5347(96)10041-0 |pmid=21237881 |bibcode=1996TEcoE..11..367Y |access-date=2023-09-06 |archive-date=2024-04-12 |archive-url=https://web.archive.org/web/20240412001342/https://linkinghub.elsevier.com/retrieve/pii/0169534796100410 |url-status=live |citeseerx=10.1.1.19.99 }} when maximum likelihood, Bayesian, or distance matrix methods are used to estimate phylogenetic trees. Phylogenetic analyzes that use the gamma distribution to model rate variation estimate a single parameter from the data because they limit consideration to distributions where {{math|1=α = λ}}. This parameterization means that the mean of this distribution is 1 and the variance is {{math|1/α}}. Maximum likelihood and Bayesian methods typically use a discrete approximation to the continuous gamma distribution.{{Cite journal |last=Yang |first=Ziheng |date=September 1994 |title=Maximum likelihood phylogenetic estimation from DNA sequences with variable rates over sites: Approximate methods |url=http://link.springer.com/10.1007/BF00160154 |journal=Journal of Molecular Evolution |language=en |volume=39 |issue=3 |pages=306–314 |doi=10.1007/BF00160154 |pmid=7932792 |bibcode=1994JMolE..39..306Y |s2cid=17911050 |issn=0022-2844 |access-date=2023-09-06 |archive-date=2024-10-09 |archive-url=https://web.archive.org/web/20241009203844/https://link.springer.com/article/10.1007/BF00160154 |url-status=live |citeseerx=10.1.1.19.6626 }}{{Cite journal |last=Felsenstein |first=Joseph |date=2001-10-01 |title=Taking Variation of Evolutionary Rates Between Sites into Account in Inferring Phylogenies |url=http://link.springer.com/10.1007/s002390010234 |journal=Journal of Molecular Evolution |volume=53 |issue=4–5 |pages=447–455 |doi=10.1007/s002390010234 |pmid=11675604 |bibcode=2001JMolE..53..447F |s2cid=9791493 |issn=0022-2844 |access-date=2023-09-06 |archive-date=2024-10-09 |archive-url=https://web.archive.org/web/20241009203754/https://link.springer.com/article/10.1007/s002390010234 |url-status=live |url-access=subscription }}

Given the scaling property above, it is enough to generate gamma variables with {{math|1=θ = 1}}, as we can later convert to any value of {{mvar|λ}} with a simple division.

Suppose we wish to generate random variables from {{math|Gamma(n + δ, 1)}}, where n is a non-negative integer and {{math|0 < δ < 1}}. Using the fact that a {{math|Gamma(1, 1)}} distribution is the same as an {{math|Exp(1)}} distribution, and noting the method of generating exponential variables, we conclude that if {{mvar|U}} is uniformly distributed on (0, 1], then {{math|−ln U}} is distributed {{math|Gamma(1, 1)}} (i.e. inverse transform sampling). Now, using the "{{mvar|α}}-addition" property of gamma distribution, we expand this result:

$$-\backslash sum\_\{k=1\}^n\; \backslash ln\; U\_k\; \backslash sim\; \backslash Gamma(n,\; 1)$$

where {{math|U_k}} are all uniformly distributed on (0, 1] and independent. All that is left now is to generate a variable distributed as {{math|Gamma(δ, 1)}} for {{math|0 < δ < 1}} and apply the "{{mvar|α}}-addition" property once more. This is the most difficult part.

Random generation of gamma variates is discussed in detail by Devroye,{{cite book |publisher=Springer-Verlag |location=New York |year=1986 |last=Devroye |first=Luc |url=http://luc.devroye.org/rnbookindex.html |title=Non-Uniform Random Variate Generation |isbn=978-0-387-96305-1 |access-date=2012-02-26 |archive-date=2012-07-17 |archive-url=https://web.archive.org/web/20120717112308/http://luc.devroye.org/rnbookindex.html |url-status=live }} See Chapter 9, Section 3.{{Rp|401–428}} noting that none are uniformly fast for all shape parameters. For small values of the shape parameter, the algorithms are often not valid.{{Rp|406}} For arbitrary values of the shape parameter, one can apply the Ahrens and Dieter{{Cite journal |last1=Ahrens |first1=J. H. |last2=Dieter |first2=U |date=January 1982 |title=Generating gamma variates by a modified rejection technique |journal=Communications of the ACM |volume=25 |issue=1 |pages=47–54 |doi=10.1145/358315.358390|s2cid=15128188 |doi-access=free }}. See Algorithm GD, p. 53. modified acceptance-rejection method Algorithm GD (shape {{math|α ≥ 1}}), or transformation method{{cite journal |last1=Ahrens |first1=J. H. |last2=Dieter |first2=U. |year=1974 |title=Computer methods for sampling from gamma, beta, Poisson and binomial distributions |journal=Computing |volume=12 |issue=3 |pages=223–246 |citeseerx=10.1.1.93.3828 |doi=10.1007/BF02293108|s2cid=37484126 }} when {{math|0 < α < 1}}. Also see Cheng and Feast Algorithm GKM 3{{cite journal |url=https://www.jstor.org/stable/2347200|jstor=2347200 |title=Some Simple Gamma Variate Generators |last1=Cheng |first1=R. C. H. |last2=Feast |first2=G. M. |journal=Journal of the Royal Statistical Society. Series C (Applied Statistics) |year=1979 |volume=28 |issue=3 |pages=290–295 |doi=10.2307/2347200 |url-access=subscription }} or Marsaglia's squeeze method.Marsaglia, G. The squeeze method for generating gamma variates. Comput, Math. Appl. 3 (1977), 321–325.

The following is a version of the Ahrens-Dieter acceptance–rejection method:

1. Generate {{mvar|U}}, {{mvar|V}} and {{mvar|W}} as iid uniform (0, 1] variates.
2. If $U\backslash le\backslash frac\; e\; \{e+\backslash delta\}$ then $\backslash xi=V^\{1/\backslash delta\}$ and $\backslash eta=W\backslash xi^\{\backslash delta-1\}$. Otherwise, $\backslash xi=1-\backslash ln\; V$ and $\backslash eta=We^\{-\backslash xi\}$.
3. If $\backslash eta>\backslash xi^\{\backslash delta-1\}e^\{-\backslash xi\}$ then go to step 1.
4. {{mvar|ξ}} is distributed as {{math|Γ(δ, 1)}}.

A summary of this is

$$\backslash theta\; \backslash left(\; \backslash xi\; -\; \backslash sum\_\{i=1\}^\{\backslash lfloor\; \backslash alpha\; \backslash rfloor\}\; \backslash ln\; U\_i\; \backslash right)\; \backslash sim\; \backslash Gamma\; (\backslash alpha,\; \backslash theta)$$

where $\backslash scriptstyle\; \backslash lfloor\; \backslash alpha\; \backslash rfloor$ is the integer part of {{mvar|α}}, {{mvar|ξ}} is generated via the algorithm above with {{math|1=δ = {{mset|α}}}} (the fractional part of {{mvar|α}}) and the {{math|U_k}} are all independent.

While the above approach is technically correct, Devroye notes that it is linear in the value of {{mvar|α}} and generally is not a good choice. Instead, he recommends using either rejection-based or table-based methods, depending on context.{{Rp|401–428}}

For example, Marsaglia's simple transformation-rejection method relying on one normal variate {{mvar|X}} and one uniform variate {{mvar|U}}:{{cite journal |last1=Marsaglia|first1=G. |last2=Tsang |first2=W. W. |year=2000 |title=A simple method for generating gamma variables|journal=ACM Transactions on Mathematical Software|volume=26 |issue=3 |pages=363–372 |doi=10.1145/358407.358414|s2cid=2634158 }}

1. Set $d\; =\; a\; -\; \backslash frac13$ and $c\; =\; \backslash frac1\{\backslash sqrt\{9d\}\}$.
2. Set $v=(1+cX)^3$.
3. If $v\; >\; 0$ and return $dv$, else go back to step 2.

With $1\; \backslash le\; a\; =\; \backslash alpha$ generates a gamma distributed random number in time that is approximately constant with {{mvar|&alpha}}. The acceptance rate does depend on {{mvar|α}}, with an acceptance rate of 0.95, 0.98, and 0.99 for α = 1, 2, and 4. For {{math|α < 1}}, one can use $\backslash gamma\_\backslash alpha\; =\; \backslash gamma\_\{1+\backslash alpha\}\; U^\{1/\backslash alpha\}$ to boost {{mvar|k}} to be usable with this method.

In Matlab numbers can be generated using the function gamrnd(), which uses the α, θ representation.

{{Reflist|30em}}

{{Wikibooks|Statistics|Distributions/Gamma|Gamma distribution}}

• {{springer|title=Gamma-distribution|id=p/g043300}}
• {{MathWorld|urlname=GammaDistribution|title=Gamma distribution}}
• ModelAssist (2017) [http://www.epixanalytics.com/modelassist/AtRisk/Model_Assist.htm#Distributions/Continuous_distributions/Gamma.htm Uses of the gamma distribution in risk modeling, including applied examples in Excel] {{Webarchive|url=https://web.archive.org/web/20170509042425/http://www.epixanalytics.com/modelassist/AtRisk/Model_Assist.htm#Distributions/Continuous_distributions/Gamma.htm |date=2017-05-09 }}.
• [http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm Engineering Statistics Handbook]

{{ProbDistributions|continuous-semi-infinite}}

{{DEFAULTSORT:Gamma Distribution}}

Category:Continuous distributions

Category:Factorial and binomial topics

Category:Conjugate prior distributions

Category:Exponential family distributions

Category:Infinitely divisible probability distributions

Category:Survival analysis

Category:Gamma and related functions