Characterization using shape α and rate β

The gamma distribution can be parameterized in terms of a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter. A random variable X that is gamma-distributed with shape α and rate β is denoted

${\displaystyle X\sim \Gamma (\alpha ,\beta )\equiv \operatorname {Gamma} (\alpha ,\beta )}$

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

${\displaystyle {\begin{aligned}f(x;\alpha ,\beta )&={\frac {x^{\alpha -1}e^{-\beta x}\beta ^{\alpha }}{\Gamma (\alpha )}}\quad {\text{ for }}x>0\quad \alpha ,\beta >0,\\[6pt]\end{aligned}}}$

where ${\displaystyle \Gamma (\alpha )}$ is the gamma function. For all positive integers, ${\displaystyle \Gamma (\alpha )=(\alpha -1)!}$.

The cumulative distribution function is the regularized gamma function:

${\displaystyle F(x;\alpha ,\beta )=\int _{0}^{x}f(u;\alpha ,\beta )\,du={\frac {\gamma (\alpha ,\beta x)}{\Gamma (\alpha )}},}$

where ${\displaystyle \gamma (\alpha ,\beta x)}$ is the lower incomplete gamma function.

If α is a positive integer (i.e., the distribution is an Erlang distribution), the cumulative distribution function has the following series expansion:^[4]

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

Characterization using shape k and scale θ

A random variable X that is gamma-distributed with shape k and scale θ is denoted by

${\displaystyle X\sim \Gamma (k,\theta )\equiv \operatorname {Gamma} (k,\theta )}$

Illustration of the gamma PDF for parameter values over k and x with θ set to 1, 2, 3, 4, 5 and 6. One can see each θ layer by itself here as well as by k and x. .

The probability density function using the shape-scale parametrization is

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

Here Γ(k) is the gamma function evaluated at k.

The cumulative distribution function is the regularized gamma function:

${\displaystyle F(x;k,\theta )=\int _{0}^{x}f(u;k,\theta )\,du={\frac {\gamma \left(k,{\frac {x}{\theta }}\right)}{\Gamma (k)}},}$

where ${\displaystyle \gamma \left(k,{\frac {x}{\theta }}\right)}$ is the lower incomplete gamma function.

It can also be expressed as follows, if k is a positive integer (i.e., the distribution is an Erlang distribution):^[4]

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

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

Mean and variance

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

${\displaystyle \mu =k\theta =\alpha /\beta }$

The variance is:

${\displaystyle \sigma ^{2}=k\theta ^{2}=\alpha /\beta ^{2}}$

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

${\displaystyle \sigma /\mu =k^{-0.5}=1/{\sqrt {\alpha }}}$

Skewness

The skewness of the gamma distribution only depends on its shape parameter, k, and it is equal to ${\displaystyle 2/{\sqrt {k}}.}$

Higher moments

The nth raw moment is given by:

${\displaystyle \mathrm {E} [X^{n}]=\theta ^{n}{\frac {\Gamma (k+n)}{\Gamma (k)}}=\theta ^{n}\prod _{i=1}^{n}(k+i-1)\;{\text{ for }}n=1,2,\ldots .}$

Median approximations and bounds

Bounds and asymptotic approximations to the median of the gamma distribution. The cyan-colored region indicates the large gap between published lower and upper bounds.

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 ${\displaystyle \nu }$ such that

${\displaystyle {\frac {1}{\Gamma (k)\theta ^{k}}}\int _{0}^{\nu }x^{k-1}e^{-x/\theta }dx={\frac {1}{2}}.}$

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 ${\displaystyle \theta =1}$)

${\displaystyle k-{\frac {1}{3}}<\nu (k)<k,}$

where ${\displaystyle \mu (k)=k}$ is the mean and ${\displaystyle \nu (k)}$ is the median of the ${\displaystyle {\text{Gamma}}(k,1)}$ distribution.^[5] For other values of the scale parameter, the mean scales to ${\displaystyle \mu =k\theta }$, and the median bounds and approximations would be similarly scaled by ${\displaystyle \theta }$.

K. P. Choi found the first five terms in a Laurent series asymptotic approximation of the median by comparing the median to Ramanujan's ${\displaystyle \theta }$ function.^[6] Berg and Pedersen found more terms:^[7]

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

Two gamma distribution median asymptotes which were proved in 2023 to be bounds (upper solid red and lower dashed red), of the from ${\displaystyle \nu (k)\approx 2^{-1/k}(A+k)}$, and an interpolation between them that makes an approximation (dotted red) that is exact at k = 1 and has maximum relative error of about 0.6%. The cyan shaded region is the remaining gap between upper and lower bounds (or conjectured bounds), including these new bounds and the bounds in the previous figure.

Log–log plot 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 conjectured bounds further narrow the yellow. Mostly within the yellow, closed-form rational-function-interpolated 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 ${\displaystyle k}$; they are not plotted in the figure, which is focused on the low-${\displaystyle k}$ region that is less well approximated.

Berg and Pedersen also proved many properties of the median, showing that it is a convex function of ${\displaystyle k}$,^[8] and that the asymptotic behavior near ${\displaystyle k=0}$ is ${\displaystyle \nu (k)\approx e^{-\gamma }2^{-1/k}}$ (where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant), and that for all ${\displaystyle k>0}$ the median is bounded by ${\displaystyle k2^{-1/k}<\nu (k)<ke^{-1/3k}}$.^[7]

A closer linear upper bound, for ${\displaystyle k\geq 1}$ only, was provided in 2021 by Gaunt and Merkle,^[9] relying on the Berg and Pedersen result that the slope of ${\displaystyle \nu (k)}$ is everywhere less than 1:

${\displaystyle \nu (k)\leq k-1+\log 2~~}$ for ${\displaystyle k\geq 1}$ (with equality at ${\displaystyle k=1}$)

which can be extended to a bound for all ${\displaystyle k>0}$ by taking the max with the chord shown in the figure, since the median was proved convex.^[8]

An approximation to the median that is asymptotically accurate at high ${\displaystyle k}$ and reasonable down to ${\displaystyle k=0.5}$ or a bit lower follows from the Wilson–Hilferty transformation:

${\displaystyle \nu (k)=k\left(1-{\frac {1}{9k}}\right)^{3}}$

which goes negative for ${\displaystyle k<1/9}$.

In 2021, Lyon proposed several approximations of the form ${\displaystyle \nu (k)\approx 2^{-1/k}(A+Bk)}$. He conjectured values of ${\displaystyle A}$ and ${\displaystyle B}$ for which this approximation is an asymptotically tight upper or lower bound for all ${\displaystyle k>0}$.^[10] In particular, he proposed these closed-form bounds, which he proved in 2023:^[11]

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

${\displaystyle \nu _{U}(k)=2^{-1/k}(e^{-\gamma }+k)\quad }$ is an upper bound, asymptotically tight as ${\displaystyle k\to 0}$

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 ${\displaystyle e^{-x}}$:

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

and the tangent line at ${\displaystyle k=1}$ where the derivative was found to be ${\displaystyle \nu ^{\prime }(1)\approx 0.9680448}$:

${\displaystyle \nu (k)\geq \nu (1)+(k-1)\nu ^{\prime }(1)\quad }$ (with equality at ${\displaystyle k=1}$)

${\displaystyle \nu (k)\geq \log(2)+(k-1)(\gamma -2\operatorname {Ei} (-\log 2)-\log \log 2)}$

where Ei is the exponential integral.^[10][11]

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

${\displaystyle \nu (k)\approx {\tilde {g}}(k)\nu _{L\infty }(k)+(1-{\tilde {g}}(k))\nu _{U}(k)}$

where ${\displaystyle {\tilde {g}}}$ is an interpolating function running monotonically from 0 at low ${\displaystyle k}$ to 1 at high ${\displaystyle k}$, approximating an ideal, or exact, interpolator ${\displaystyle g(k)}$:

${\displaystyle g(k)={\frac {\nu _{U}(k)-\nu (k)}{\nu _{U}(k)-\nu _{L\infty }(k)}}}$

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

${\displaystyle {\tilde {g}}_{1}(k)={\frac {k}{b_{0}+k}}}$

the tightest lower bound has

${\displaystyle b_{0}={\frac {{\frac {8}{405}}+e^{-\gamma }\log 2-{\frac {\log ^{2}2}{2}}}{e^{-\gamma }-\log 2+{\frac {1}{3}}}}-\log 2\approx 0.143472}$

and the tightest upper bound has

${\displaystyle b_{0}={\frac {e^{-\gamma }-\log 2+{\frac {1}{3}}}{1-{\frac {e^{-\gamma }\pi ^{2}}{12}}}}\approx 0.374654}$

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.^[10]

Summation

If X_i has a Gamma(k_i, θ) distribution for i = 1, 2, ..., N (i.e., all distributions have the same scale parameter θ), then

${\displaystyle \sum _{i=1}^{N}X_{i}\sim \mathrm {Gamma} \left(\sum _{i=1}^{N}k_{i},\theta \right)}$

provided all X_i are independent.

For the cases where the X_i are independent but have different scale parameters, see Mathai ^[12] or Moschopoulos.^[13]

The gamma distribution exhibits infinite divisibility.

Scaling

If

${\displaystyle X\sim \mathrm {Gamma} (k,\theta ),}$

then, for any c > 0,

${\displaystyle cX\sim \mathrm {Gamma} (k,c\,\theta ),}$ by moment generating functions,

or equivalently, if

${\displaystyle X\sim \mathrm {Gamma} \left(\alpha ,\beta \right)}$ (shape-rate parameterization)

${\displaystyle cX\sim \mathrm {Gamma} \left(\alpha ,{\frac {\beta }{c}}\right),}$

Indeed, we know that if X is an exponential r.v. with rate λ, then cX is an exponential r.v. with rate λ/c; the same thing is valid with Gamma variates (and this can be checked using the moment-generating function, see, e.g.,these notes, 10.4-(ii)): multiplication by a positive constant c divides the rate (or, equivalently, multiplies the scale).

Exponential family

The gamma distribution is a two-parameter exponential family with natural parameters k − 1 and −1/θ (equivalently, α − 1 and −β), and natural statistics X and ln(X).

If the shape parameter k is held fixed, the resulting one-parameter family of distributions is a natural exponential family.

Logarithmic expectation and variance

One can show that

${\displaystyle \operatorname {E} [\ln(X)]=\psi (\alpha )-\ln(\beta )}$

or equivalently,

${\displaystyle \operatorname {E} [\ln(X)]=\psi (k)+\ln(\theta )}$

where ${\displaystyle \psi }$ is the digamma function. Likewise,

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

where ${\displaystyle \psi ^{(1)}}$ 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 ln(x).

Information entropy

The information entropy is

${\displaystyle {\begin{aligned}\operatorname {H} (X)&=\operatorname {E} [-\ln(p(X))]\\[4pt]&=\operatorname {E} [-\alpha \ln(\beta )+\ln(\Gamma (\alpha ))-(\alpha -1)\ln(X)+\beta X]\\[4pt]&=\alpha -\ln(\beta )+\ln(\Gamma (\alpha ))+(1-\alpha )\psi (\alpha ).\end{aligned}}}$

In the k, θ parameterization, the information entropy is given by

${\displaystyle \operatorname {H} (X)=k+\ln(\theta )+\ln(\Gamma (k))+(1-k)\psi (k).}$

Kullback–Leibler divergence

Illustration of the Kullback–Leibler (KL) divergence for two gamma PDFs. Here β = β₀ + 1 which are set to 1, 2, 3, 4, 5 and 6. The typical asymmetry for the KL divergence is clearly visible.

The Kullback–Leibler divergence (KL-divergence), of Gamma(α_p, β_p) ("true" distribution) from Gamma(α_q, β_q) ("approximating" distribution) is given by^[14]

${\displaystyle {\begin{aligned}D_{\mathrm {KL} }(\alpha _{p},\beta _{p};\alpha _{q},\beta _{q})={}&(\alpha _{p}-\alpha _{q})\psi (\alpha _{p})-\log \Gamma (\alpha _{p})+\log \Gamma (\alpha _{q})\\&{}+\alpha _{q}(\log \beta _{p}-\log \beta _{q})+\alpha _{p}{\frac {\beta _{q}-\beta _{p}}{\beta _{p}}}.\end{aligned}}}$

Written using the k, θ parameterization, the KL-divergence of Gamma(k_p, θ_p) from Gamma(k_q, θ_q) is given by

${\displaystyle {\begin{aligned}D_{\mathrm {KL} }(k_{p},\theta _{p};k_{q},\theta _{q})={}&(k_{p}-k_{q})\psi (k_{p})-\log \Gamma (k_{p})+\log \Gamma (k_{q})\\&{}+k_{q}(\log \theta _{q}-\log \theta _{p})+k_{p}{\frac {\theta _{p}-\theta _{q}}{\theta _{q}}}.\end{aligned}}}$

Laplace transform

The Laplace transform of the gamma PDF is

${\displaystyle F(s)=(1+\theta s)^{-k}={\frac {\beta ^{\alpha }}{(s+\beta )^{\alpha }}}.}$

General

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

${\displaystyle X\sim \Gamma (k\in \mathbf {Z} ,\theta ),\qquad Y\sim \operatorname {Pois} \left({\frac {x}{\theta }}\right),}$

then

${\displaystyle P(X>x)=P(Y<k).}$

• If X has a Maxwell–Boltzmann distribution with parameter a, then

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

• If X ~ Gamma(k, θ), then ${\textstyle \log X}$ follows an exponential-gamma (abbreviated exp-gamma) distribution.^[15] It is sometimes referred to as the log-gamma distribution.^[16] Formulas for its mean and variance are in the section #Logarithmic expectation and variance.
• If X ~ Gamma(k, θ), then ${\displaystyle {\sqrt {X}}}$ follows a generalized gamma distribution with parameters p = 2, d = 2k, and ${\displaystyle a={\sqrt {\theta }}}$ .
• More generally, if X ~ Gamma(k,θ), then ${\displaystyle X^{q}}$ for ${\displaystyle q>0}$ follows a generalized gamma distribution with parameters p = 1/q, d = k/q, and ${\displaystyle a=\theta ^{q}}$.
• If X ~ Gamma(k, θ) with shape k and scale θ, then 1/X ~ Inv-Gamma(k, θ⁻¹) (see Inverse-gamma distribution for derivation).
• Parametrization 1: If ${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})\,}$ are independent, then ${\displaystyle {\frac {\alpha _{2}\theta _{2}X_{1}}{\alpha _{1}\theta _{1}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}$, or equivalently, ${\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\theta _{1}}{\theta _{2}}}\right)}$
• Parametrization 2: If ${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})\,}$ are independent, then ${\displaystyle {\frac {\alpha _{2}\beta _{1}X_{1}}{\alpha _{1}\beta _{2}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}$, or equivalently, ${\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\beta _{2}}{\beta _{1}}}\right)}$
• If X ~ Gamma(α, θ) and Y ~ Gamma(β, θ) are independently distributed, then X/(X + Y) has a beta distribution with parameters α and β, and X/(X + Y) is independent of X + Y, which is Gamma(α + β, θ)-distributed.
• If X_i ~ Gamma(α_i, 1) are independently distributed, then the vector (X₁/S, ..., X_n/S), where S = X₁ + ... + X_n, follows a Dirichlet distribution with parameters α₁, ..., α_n.
• For large k the gamma distribution converges to normal distribution with mean μ = kθ and variance σ² = kθ².
• The gamma distribution is the conjugate prior for the precision of the normal distribution with known mean.
• The matrix gamma distribution and the Wishart distribution are multivariate generalizations of the gamma distribution (samples are positive-definite matrices rather than positive real numbers).
• The gamma distribution is a special case of the generalized gamma distribution, the generalized integer gamma distribution, and the generalized inverse Gaussian distribution.
• Among the discrete distributions, the negative binomial distribution is sometimes considered the discrete analog of the gamma distribution.
• Tweedie distributions – the gamma distribution is a member of the family of Tweedie exponential dispersion models.
• Modified Half-normal distribution – the Gamma distribution is a member of the family of Modified half-normal distribution.^[17] The corresponding density is ${\displaystyle f(x\mid \alpha ,\beta ,\gamma )={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}$, where ${\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}$ denotes the Fox–Wright Psi function.
• For the shape-scale parameterization ${\displaystyle x|\theta \sim \Gamma (k,\theta )}$, if the scale parameter ${\displaystyle \theta \sim IG(b,1)}$ where ${\displaystyle IG}$ denotes the Inverse-gamma distribution, then the marginal distribution ${\displaystyle x\sim \beta '(k,b)}$ where ${\displaystyle \beta '}$ denotes the Beta prime distribution.

Compound gamma

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.^[18]

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.

Weibull and stable count

The gamma distribution ${\displaystyle f(x;k)\,(k>1)}$ can be expressed as the product distribution of a Weibull distribution and a variant form of the stable count distribution. Its shape parameter ${\displaystyle k}$ can be regarded as the inverse of Lévy's stability parameter in the stable count distribution:

$${\displaystyle f(x;k)=\displaystyle \int _{0}^{\infty }{\frac {1}{u}}\,W_{k}\left({\frac {x}{u}}\right)\left[ku^{k-1}\,{\mathfrak {N}}_{\frac {1}{k}}\left(u^{k}\right)\right]\,du,}$$

where ${\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}$ is a standard stable count distribution of shape ${\displaystyle \alpha =1/k}$, and ${\displaystyle W_{k}(x)}$ is a standard Weibull distribution of shape ${\displaystyle k}$.

Parameter estimation

Bayesian inference