{"history":[],"article":{"id":"93d887ff-4b22-4021-904b-9f1f31fab1ab","title":"Gamma distribution","slug":"gamma-distribution","content":":::summary\n**Name:** Gamma \n**Type:** density \n**Pdf Image:** [[Image:Gamma distribution pdf.svg|325px|Probability density plots of gamma distributions]] \n**Cdf Image:** [[Image:Gamma distribution cdf.svg|325px|Cumulative distribution plots of gamma distributions]] \n**Parameters:** k > 0 · [shape](https://bliptext.com/articles/shape-parameter) · θ > 0 · [scale](https://bliptext.com/articles/scale-parameter) \n**Mean:** k \\theta \n**Median:** No simple closed form \n**Mode:** , \n**Variance:** k \\theta^2 \n**Kurtosis:** \\frac{6}{k} \n**Parameters 2:** • α > 0 · shape · • β > 0 · rate \n**Median 2:** No simple closed form \n:::\n\n# \nIn [probability theory](https://bliptext.com/articles/probability-theory) and [statistics](https://bliptext.com/articles/statistics), the **gamma distribution** is a versatile two-[parameter](https://bliptext.com/articles/statistical-parameter) family of continuous [probability distributions](https://bliptext.com/articles/probability-distribution).\nThe [exponential distribution](https://bliptext.com/articles/exponential-distribution), [Erlang distribution](https://bliptext.com/articles/erlang-distribution), and [chi-squared distribution](https://bliptext.com/articles/chi-squared-distribution) are special cases of the gamma distribution.\nThere are two equivalent parameterizations in common use:\nIn each of these forms, both parameters are positive real numbers.\nThe distribution has important applications in various fields, including [econometrics](https://bliptext.com/articles/econometrics), [Bayesian statistics](https://bliptext.com/articles/bayesian-statistics), life testing.\nIn econometrics, the (k, θ) parameterization is common for modeling waiting times, such as the time until death, where it often takes the form of an [Erlang distribution](https://bliptext.com/articles/erlang-distribution) for integer k values.\nBayesian statistics prefer the (α, β) parameterization, utilizing the gamma distribution as a [conjugate prior](https://bliptext.com/articles/conjugate-prior) for several inverse scale parameters, facilitating analytical tractability in posterior distribution computations.\nThe 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.\nThe 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.\nIts mathematical properties, such as mean, variance, skewness, and higher moments, provide a toolset for statistical analysis and inference.\nPractical applications of the distribution span several disciplines, underscoring its importance in theoretical and applied statistics.\nThe gamma distribution is the [maximum entropy probability distribution](https://bliptext.com/articles/maximum-entropy-probability-distribution) (both with respect to a uniform base measure and a 1/x base measure) for a random variable X for which\nE[X] = kθ = α/β\nis fixed and greater than zero, and\nE[ln X] = ψ(k) + ln θ = ψ(α) − ln β\nis fixed (ψ is the [digamma function](https://bliptext.com/articles/digamma-function)).\n\n# Definitions\nThe parameterization with k and θ appears to be more common in [econometrics](https://bliptext.com/articles/econometrics) and other applied fields, where the gamma distribution is frequently used to model waiting times.\nFor instance, in [life testing](https://bliptext.com/articles/accelerated-life-testing), the waiting time until death is a [random variable](https://bliptext.com/articles/random-variable) that is frequently modeled with a gamma distribution.\nSee Hogg and Craig for an explicit motivation.\nThe parameterization with α and β is more common in [Bayesian statistics](https://bliptext.com/articles/bayesian-statistics), where the gamma distribution is used as a [conjugate prior](https://bliptext.com/articles/conjugate-prior) distribution for various types of inverse scale (rate) parameters, such as the λ of an [exponential distribution](https://bliptext.com/articles/exponential-distribution) or a [Poisson distribution](https://bliptext.com/articles/poisson-distribution) – or for that matter, the β of the gamma distribution itself.\nThe closely related [inverse-gamma distribution](https://bliptext.com/articles/inverse-gamma-distribution) is used as a conjugate prior for scale parameters, such as the [variance](https://bliptext.com/articles/variance) of a [normal distribution](https://bliptext.com/articles/normal-distribution).\nIf k is a positive [integer](https://bliptext.com/articles/integer), then the distribution represents an [Erlang distribution](https://bliptext.com/articles/erlang-distribution); i.e., the sum of k independent [exponentially distributed](https://bliptext.com/articles/exponentially-distributed) [random variables](https://bliptext.com/articles/random-variable), each of which has a mean of θ.\n\n## Characterization using shape α and rate β\nThe gamma distribution can be parameterized in terms of a [shape parameter](https://bliptext.com/articles/shape-parameter)\nα = k\nand an inverse scale parameter\nβ = 1/θ\n, called a [rate parameter](https://bliptext.com/articles/rate-parameter).\nA random variable X that is gamma-distributed with shape α and rate β is denoted\nThe corresponding probability density function in the shape-rate parameterization is\nwhere is the [gamma function](https://bliptext.com/articles/gamma-function).\nFor all positive integers,.\nThe [cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function) is the regularized gamma function:\nwhere is the lower [incomplete gamma function](https://bliptext.com/articles/incomplete-gamma-function).\nIf α is a positive [integer](https://bliptext.com/articles/integer) (i.e., the distribution is an [Erlang distribution](https://bliptext.com/articles/erlang-distribution)), the cumulative distribution function has the following series expansion:\n\n## Characterization using shape k and scale θ\nA random variable X that is gamma-distributed with shape k and scale θ is denoted by\n[[Image:Gamma-PDF-3D.png|thumb|right|320px|Illustration of the gamma PDF for parameter values over k and x with θ set to\n1, 2, 3, 4, 5,\nand\n6\n. One can see each θ layer by itself here as well as by k and x. .]]\nThe [probability density function](https://bliptext.com/articles/probability-density-function) using the shape-scale parametrization is\nHere\nΓ(k)\nis the [gamma function](https://bliptext.com/articles/gamma-function) evaluated at k.\nThe [cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function) is the regularized gamma function:\nwhere is the lower [incomplete gamma function](https://bliptext.com/articles/incomplete-gamma-function).\nIt can also be expressed as follows, if k is a positive [integer](https://bliptext.com/articles/integer) (i.e., the distribution is an [Erlang distribution](https://bliptext.com/articles/erlang-distribution)):\nBoth parametrizations are common because either can be more convenient depending on the situation.\n# [gamma distribution | upload.wikimedia.org/wikipedia/commons/7/78/Gamma-PDF-3D-by-k.png]\n# [gamma distribution | upload.wikimedia.org/wikipedia/commons/9/91/Gamma-PDF-3D-by-Theta.png]\n# [gamma distribution | upload.wikimedia.org/wikipedia/commons/c/cb/Gamma-PDF-3D-by-x.png]\n\n# Properties\n\n## Mean and variance\nThe mean of gamma distribution is given by the product of its shape and scale parameters:\nThe variance is:\nThe square root of the inverse shape parameter gives the [coefficient of variation](https://bliptext.com/articles/coefficient-of-variation):\n\n## Skewness\nThe [skewness](https://bliptext.com/articles/skewness) of the gamma distribution only depends on its shape parameter, k, and it is equal to 2/\\sqrt{k}.\n\n## Higher moments\nThe n-th [raw moment](https://bliptext.com/articles/raw-moment) is given by:\n\n## Median approximations and bounds\nUnlike the mode and the mean, which have readily calculable formulas based on the parameters, the median does not have a closed-form equation.\nThe median for this distribution is the value \\nu such that\nA 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 \\theta = 1)\nwhere \\mu(k) = k is the mean and \\nu(k) is the median of the distribution.\nFor other values of the scale parameter, the mean scales to, and the median bounds and approximations would be similarly scaled by θ.\nK. P. Choi found the first five terms in a [Laurent series](https://bliptext.com/articles/laurent-series) asymptotic approximation of the median by comparing the median to [Ramanujan's \\theta function](https://bliptext.com/articles/ramanujan-theta-function).\nBerg and Pedersen found more terms:\nPartial sums of these series are good approximations for high enough k; they are not plotted in the figure, which is focused on the low-k region that is less well approximated.\nBerg and Pedersen also proved many properties of the median, showing that it is a convex function of k, and that the asymptotic behavior near k = 0 is (where γ is the [Euler–Mascheroni constant](https://bliptext.com/articles/euler-mascheroni-constant)), and that for all k > 0 the median is bounded by.\nA closer linear upper bound, for k \\ge 1 only, was provided in 2021 by Gaunt and Merkle, relying on the Berg and Pedersen result that the slope of \\nu(k) is everywhere less than 1:\nfor k \\ge 1 (with equality at k = 1)\nwhich can be extended to a bound for all k > 0 by taking the max with the chord shown in the figure, since the median was proved convex.\nAn approximation to the median that is asymptotically accurate at high k and reasonable down to k = 0.5 or a bit lower follows from the [Wilson–Hilferty transformation](https://bliptext.com/articles/wilson-hilferty-transformation):\nwhich goes negative for k < 1/9.\nIn 2021, Lyon proposed several approximations of the form.\nHe conjectured values of A and B for which this approximation is an asymptotically tight upper or lower bound for all k > 0.\nIn particular, he proposed these closed-form bounds, which he proved in 2023:\nis a lower bound, asymptotically tight as is an upper bound, asymptotically tight as k \\to 0\nLyon also showed (informally in 2021, rigorously in 2023) two other lower bounds that are not [closed-form expressions](https://bliptext.com/articles/closed-form-expression), including this one involving the [gamma function](https://bliptext.com/articles/gamma-function), based on solving the integral expression substituting 1 for e^{-x}:\n(approaching equality as k \\to 0)\nand the tangent line at k = 1 where the derivative was found to be :\n(with equality at k = 1)\nwhere Ei is the [exponential integral](https://bliptext.com/articles/exponential-integral).\nAdditionally, he showed that interpolations between bounds could provide excellent approximations or tighter bounds to the median, including an approximation that is exact at k = 1 (where ) and has a maximum relative error less than 0.6%.\nInterpolated approximations and bounds are all of the form\nwhere \\tilde{g} is an interpolating function running monotonically from 0 at low k to 1 at high k, approximating an ideal, or exact, interpolator g(k):\nFor the simplest interpolating function considered, a first-order rational function\nthe tightest lower bound has\nand the tightest upper bound has\nThe interpolated bounds are plotted (mostly inside the yellow region) in the [log–log plot](https://bliptext.com/articles/log-log-plot) shown.\nEven tighter bounds are available using different interpolating functions, but not usually with closed-form parameters like these.\n# [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 before 2021. | upload.wikimedia.org/wikipedia/commons/4/42/Gamma///distribution///median///bounds.png]\n# [Two gamma distribution median asymptotes which were proved in 2023 to be bounds (upper solid red and lower dashed red), of the from, 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. | upload.wikimedia.org/wikipedia/commons/1/1e/Gamma///distribution///median///Lyon///bounds.png]\n# [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 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. | upload.wikimedia.org/wikipedia/commons/a/a7/Gamma///distribution///median///loglog///bounds.png]\n\n## Summation\nIf\nXi\nhas a\nGamma(ki, θ)\ndistribution for\ni = 1, 2, ..., N\n(i.e., all distributions have the same scale parameter θ), then\nprovided all\nXi\nare [independent](https://bliptext.com/articles/statistical-independence).\nFor the cases where the\nXi\nare [independent](https://bliptext.com/articles/statistical-independence) but have different scale parameters, see Mathai or Moschopoulos.\nThe gamma distribution exhibits [infinite divisibility](https://bliptext.com/articles/infinite-divisibility-probability).\n\n## Scaling\nIf\nthen, for any\nc > 0\n,\nby moment generating functions,\nor equivalently, if\n(shape-rate parameterization)\nIndeed, we know that if X is an [exponential r.v.](https://bliptext.com/articles/exponential-distribution) with rate λ, then\ncX\nis an exponential r.v. with rate\nλ/c\n\n## Exponential family\nThe gamma distribution is a two-parameter [exponential family](https://bliptext.com/articles/exponential-family) with [natural parameters](https://bliptext.com/articles/natural-parameters)\nk − 1\nand\n−1/θ\n(equivalently,\nα − 1\nand\n−β\n), and [natural statistics](https://bliptext.com/articles/natural-statistics) X and\nln X\n.\nIf the shape parameter k is held fixed, the resulting one-parameter family of distributions is a [natural exponential family](https://bliptext.com/articles/natural-exponential-family).\n\n## Logarithmic expectation and variance\nOne can show that\nor equivalently,\nwhere ψ is the [digamma function](https://bliptext.com/articles/digamma-function).\nLikewise,\nwhere \\psi^{(1)} is the [trigamma function](https://bliptext.com/articles/trigamma-function).\nThis can be derived using the [exponential family](https://bliptext.com/articles/exponential-family) formula for the [moment generating function of the sufficient statistic](https://bliptext.com/articles/exponential-family), because one of the sufficient statistics of the gamma distribution is\nln x\n.\n\n## Information entropy\nThe [information entropy](https://bliptext.com/articles/information-entropy) is\nIn the k, θ parameterization, the [information entropy](https://bliptext.com/articles/information-entropy) is given by\n\n## Kullback–Leibler divergence\n[[Image:Gamma-KL-3D.png|thumb|right|320px|Illustration of the Kullback–Leibler (KL) divergence for two gamma PDFs. Here\nβ = β0 + 1\nwhich are set to\n1, 2, 3, 4, 5,\nand\n6\n. The typical asymmetry for the KL divergence is clearly visible.]]\nThe [Kullback–Leibler divergence](https://bliptext.com/articles/kullback-leibler-divergence) (KL-divergence), of\nGamma(αp, βp)\n(\"true\" distribution) from\nGamma(αq, βq)\n(\"approximating\" distribution) is given by\nWritten using the k, θ parameterization, the KL-divergence of\nGamma(kp, θp)\nfrom\nGamma(kq, θq)\nis given by\n\n## Laplace transform\nThe [Laplace transform](https://bliptext.com/articles/laplace-transform) of the gamma PDF is\n\n# Related distributions\n\n## General\nX ~ Gamma(1, λ)\n(in the shape–rate parametrization), then X has an [exponential distribution](https://bliptext.com/articles/exponential-distribution) with rate parameter λ.\nIn the shape-scale parametrization,\nX ~ Gamma(1, λ)\nhas an exponential distribution with rate parameter\n1/λ\n.\nX ~ Gamma(ν/2, 2)\n(in the shape–scale parametrization), then X is identical to\nχ2(ν)\n, the [chi-squared distribution](https://bliptext.com/articles/chi-squared-distribution) with ν degrees of freedom.\nConversely, if\nQ ~ χ2(ν)\nand c is a positive constant, then\ncQ ~ Gamma(ν/2, 2c)\n.\nθ = 1/k\n, one obtains the [Schulz-Zimm distribution](https://bliptext.com/articles/schulz-zimm-distribution), which is most prominently used to model polymer chain lengths.\n1/θ\n. If\nX ~ Gamma(k, θ)\n, then \\log X follows an exponential-gamma (abbreviated exp-gamma) distribution.\nIt is sometimes referred to as the log-gamma distribution.\n[](https://bliptext.com/articles/)F[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)r[](https://bliptext.com/articles/)m[](https://bliptext.com/articles/)u[](https://bliptext.com/articles/)l[](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)s[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)f[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)r[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)i[](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)s[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)m[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/)d[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)v[](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)r[](https://bliptext.com/articles/)i[](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/)c[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)r[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)i[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)h[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)s[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/)c[](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)i[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/).[](https://bliptext.com/articles/)\nX ~ Gamma(k, θ)\n, then \\sqrt{X} follows a [generalized gamma distribution](https://bliptext.com/articles/generalized-gamma-distribution) with parameters\np = 2\n,\nd = 2k\n, and.\nX ~ Gamma(k,θ)\n, then X^q for q > 0 follows a [generalized gamma distribution](https://bliptext.com/articles/generalized-gamma-distribution) with parameters\np = 1/q\n,\nd = k/q\n, and.\nX ~ Gamma(k, θ)\nwith shape k and scale θ, then\n1/X ~ Inv-Gamma(k, θ−1)\n(see [Inverse-gamma distribution](https://bliptext.com/articles/inverse-gamma-distribution) for derivation).\nX ~ Gamma(α, θ)\nand\nY ~ Gamma(β, θ)\nare independently distributed, then\nX/(X + Y)\nhas a [beta distribution](https://bliptext.com/articles/beta-distribution) with parameters α and β, and\nX/(X + Y)\nis independent of\nX + Y\n, which is\nGamma(α + β, θ)\n-distributed.\nXi ~ Gamma(αi, 1)\nare independently distributed, then the vector (\nX1/S, ..., Xn/S)\n, where\nS = X1 + ...\n+ Xn\n, follows a [Dirichlet distribution](https://bliptext.com/articles/dirichlet-distribution) with parameters\nα1, ..., αn\n.\nμ = kθ\nand variance\nσ2 = kθ2\n.\n\n## Compound gamma\nIf 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.\nThe [compound distribution](https://bliptext.com/articles/compound-distribution), which results from integrating out the inverse scale, has a closed-form solution known as the [compound gamma distribution](https://bliptext.com/articles/compound-gamma-distribution).\nIf, 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](https://bliptext.com/articles/k-distribution).\n\n## Weibull and stable count\nThe gamma distribution can be expressed as the product distribution of a [Weibull distribution](https://bliptext.com/articles/weibull-distribution) and a variant form of the [stable count distribution](https://bliptext.com/articles/stable-count-distribution).\nIts shape parameter k can be regarded as the inverse of Lévy's stability parameter in the stable count distribution:\nwhere is a standard stable count distribution of shape, and W_k(x) is a standard Weibull distribution of shape k.\n\n# Statistical inference\n\n## Parameter estimation\n\n### Maximum likelihood estimation\nThe likelihood function for N [iid](https://bliptext.com/articles/iid) observations\n(x1, ..., xN)\nis\nfrom which we calculate the log-likelihood function\nFinding the maximum with respect to θ by taking the derivative and setting it equal to zero yields the [maximum likelihood](https://bliptext.com/articles/maximum-likelihood) estimator of the θ parameter, which equals the [sample mean](https://bliptext.com/articles/sample-mean) \\bar{x} divided by the shape parameter k:\nSubstituting this into the log-likelihood function gives\nWe need at least two samples: N\\ge2, because for N=1, the function \\ell(k) increases without bounds as k\\to\\infty.\nFor k>0, it can be verified that \\ell(k) is strictly [concave](https://bliptext.com/articles/concave-function), by using [inequality properties of the polygamma function](https://bliptext.com/articles/polygamma-function).\nFinding the maximum with respect to k by taking the derivative and setting it equal to zero yields\nwhere ψ is the [digamma function](https://bliptext.com/articles/digamma-function) and is the sample mean of\nln x\n. There is no closed-form solution for k. The function is numerically very well behaved, so if a numerical solution is desired, it can be found using, for example, [Newton's method](https://bliptext.com/articles/newton-s-method).\nAn initial value of k can be found either using the [method of moments](https://bliptext.com/articles/method-of-moments-statistics), or using the approximation\nIf we let\nthen k is approximately\nwhich is within 1.5% of the correct value.\nAn explicit form for the Newton–Raphson update of this initial guess is:\nAt the maximum-likelihood estimate, the expected values for x and \\ln x agree with the empirical averages:\n\n#### Caveat for small shape parameter\nFor data,, that is represented in a [floating point](https://bliptext.com/articles/floating-point) format that underflows to 0 for values smaller than \\varepsilon, the logarithms that are needed for the maximum-likelihood estimate will cause failure if there are any underflows.\nIf we assume the data was generated by a gamma distribution with cdf, then the probability that there is at least one underflow is:\nThis probability will approach 1 for small k and large N. For example, at k=10^{-2}, N=10^4 and,.\nA workaround is to instead have the data in logarithmic format.\nIn 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 k<1.\nFollowing the implementation in, this can be done as follows: sample and independently.\nThen the required logarithmic sample is, so that.\n\n### Closed-form estimators\nThere exist consistent closed-form estimators of k and θ that are derived from the likelihood of the [generalized gamma distribution](https://bliptext.com/articles/generalized-gamma-distribution).\nThe estimate for the shape k is\nand the estimate for the scale θ is\nUsing the sample mean of x, the sample mean of\nln x\n, and the sample mean of the product\nx·ln x\nsimplifies the expressions to:\nIf the rate parameterization is used, the estimate of.\nThese estimators are not strictly maximum likelihood estimators, but are instead referred to as mixed type log-moment estimators.\nThey have however similar efficiency as the maximum likelihood estimators.\nAlthough these estimators are consistent, they have a small bias.\nA bias-corrected variant of the estimator for the scale θ is\nA bias correction for the shape parameter k is given as\n\n### Bayesian minimum mean squared error\nWith known k and unknown θ, the posterior density function for theta (using the standard scale-invariant [prior](https://bliptext.com/articles/prior-probability) for θ) is\nDenoting\nIntegration with respect to θ can be carried out using a change of variables, revealing that\n1/θ\nis gamma-distributed with parameters\nα = Nk\n,\nβ = y\n.\nThe moments can be computed by taking the ratio (m by\nm = 0\n)\nwhich shows that the mean ± standard deviation estimate of the posterior distribution for θ is\n\n## Bayesian inference\n\n### Conjugate prior\nIn [Bayesian inference](https://bliptext.com/articles/bayesian-inference), the **gamma distribution** is the [conjugate prior](https://bliptext.com/articles/conjugate-prior) to many likelihood distributions: the [Poisson](https://bliptext.com/articles/poisson-distribution), [exponential](https://bliptext.com/articles/exponential-distribution), [normal](https://bliptext.com/articles/normal-distribution) (with known mean), [Pareto](https://bliptext.com/articles/pareto-distribution), gamma with known shape σ, [inverse gamma](https://bliptext.com/articles/inverse-gamma) with known shape parameter, and [Gompertz](https://bliptext.com/articles/gompertz-distribution) with known scale parameter.\nThe gamma distribution's [conjugate prior](https://bliptext.com/articles/conjugate-prior) is:\nwhere Z is the normalizing constant with no closed-form solution.\nThe posterior distribution can be found by updating the parameters as follows:\nwhere n is the number of observations, and\nxi\nis the i-th observation.\n\n# Occurrence and applications\nConsider a sequence of events, with the waiting time for each event being an exponential distribution with rate β.\nThen the waiting time for the n-th event to occur is the gamma distribution with integer shape \\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.\nExamples include the waiting time of [cell-division events](https://bliptext.com/articles/cell-division), number of compensatory mutations for a given mutation, waiting time until a repair is necessary for a hydraulic system, and so on.\nIn biophysics, the dwell time between steps of a molecular motor like [ATP synthase](https://bliptext.com/articles/atp-synthase) is nearly exponential at constant ATP concentration, revealing that each step of the motor takes a single ATP hydrolysis.\nIf there were n ATP hydrolysis events, then it would be a gamma distribution with degree n.\nThe gamma distribution has been used to model the size of [insurance claims](https://bliptext.com/articles/insurance-policy) and rainfalls.\nThis means that aggregate insurance claims and the amount of rainfall accumulated in a reservoir are modelled by a [gamma process](https://bliptext.com/articles/gamma-process) – much like the [exponential distribution](https://bliptext.com/articles/exponential-distribution) generates a [Poisson process](https://bliptext.com/articles/poisson-process).\nThe gamma distribution is also used to model errors in multi-level [Poisson regression](https://bliptext.com/articles/poisson-regression) models because a [mixture](https://bliptext.com/articles/mixture-distribution) of [Poisson distributions](https://bliptext.com/articles/poisson-distribution) with gamma-distributed rates has a known closed form distribution, called [negative binomial](https://bliptext.com/articles/negative-binomial).\nIn wireless communication, the gamma distribution is used to model the [multi-path fading](https://bliptext.com/articles/multi-path-fading) of signal power; see also [Rayleigh distribution](https://bliptext.com/articles/rayleigh-distribution) and [Rician distribution](https://bliptext.com/articles/rician-distribution).\nIn [oncology](https://bliptext.com/articles/oncology), the age distribution of [cancer](https://bliptext.com/articles/cancer) [incidence](https://bliptext.com/articles/disease-incidence) often follows the gamma distribution, wherein the shape and scale parameters predict, respectively, the number of [driver events](https://bliptext.com/articles/carcinogenesis) and the time interval between them.\nIn [neuroscience](https://bliptext.com/articles/neuroscience), the gamma distribution is often used to describe the distribution of [inter-spike intervals](https://bliptext.com/articles/temporal-coding).\nIn [bacterial](https://bliptext.com/articles/bacterial-genetics) [gene expression](https://bliptext.com/articles/gene-expression), the [copy number](https://bliptext.com/articles/copy-number-analysis) of a [constitutively expressed](https://bliptext.com/articles/constitutively-expressed) protein often follows the gamma distribution, where the scale and shape parameter are, respectively, the mean number of bursts per cell cycle and the mean number of [protein molecules](https://bliptext.com/articles/protein-molecule) produced by a single mRNA during its lifetime.\nIn [genomics](https://bliptext.com/articles/genomics), the gamma distribution was applied in [peak calling](https://bliptext.com/articles/peak-calling) step (i.e., in recognition of signal) in [ChIP-chip](https://bliptext.com/articles/chip-chip) and [ChIP-seq](https://bliptext.com/articles/chip-seq) data analysis.\nIn Bayesian statistics, the gamma distribution is widely used as a [conjugate prior](https://bliptext.com/articles/conjugate-prior).\nIt is the conjugate prior for the [precision](https://bliptext.com/articles/precision-statistics) (i.e. inverse of the variance) of a [normal distribution](https://bliptext.com/articles/normal-distribution).\nIt is also the conjugate prior for the [exponential distribution](https://bliptext.com/articles/exponential-distribution).\nIn [phylogenetics](https://bliptext.com/articles/phylogenetics), the gamma distribution is the most commonly used approach to model among-sites rate variation when [maximum likelihood](https://bliptext.com/articles/computational-phylogenetics), [Bayesian](https://bliptext.com/articles/bayesian-inference-in-phylogeny), or [distance matrix methods](https://bliptext.com/articles/distance-matrices-in-phylogeny) are used to estimate phylogenetic trees.\nPhylogenetic analyzes that use the gamma distribution to model rate variation estimate a single parameter from the data because they limit consideration to distributions where\nα = β\n. This parameterization means that the mean of this distribution is 1 and the variance is\n1/α\n. Maximum likelihood and Bayesian methods typically use a discrete approximation to the continuous gamma distribution.\n\n# Random variate generation\nGiven the scaling property above, it is enough to generate gamma variables with\nθ = 1\n, as we can later convert to any value of β with a simple division.\nSuppose we wish to generate random variables from\nGamma(n + δ, 1)\n, where n is a non-negative integer and\n0 < δ < 1\n. Using the fact that a\nGamma(1, 1)\ndistribution is the same as an\nExp(1)\ndistribution, and noting the method of [generating exponential variables](https://bliptext.com/articles/exponential-distribution), we conclude that if U is [uniformly distributed](https://bliptext.com/articles/uniform-distribution-continuous) on (0, 1], then\n−ln U\nis distributed\nGamma(1, 1)\n(i.e. [inverse transform sampling](https://bliptext.com/articles/inverse-transform-sampling)).\nNow, using the \"α-addition\" property of gamma distribution, we expand this result:\nwhere\nUk\nare all uniformly distributed on (0, 1] and [independent](https://bliptext.com/articles/statistical-independence). All that is left now is to generate a variable distributed as\nGamma(δ, 1)\nfor\n0 < δ < 1\nand apply the \"α-addition\" property once more.\nThis is the most difficult part.\nRandom generation of gamma variates is discussed in detail by Devroye, noting that none are uniformly fast for all shape parameters.\nFor small values of the shape parameter, the algorithms are often not valid.\nFor arbitrary values of the shape parameter, one can apply the Ahrens and Dieter modified acceptance-rejection method Algorithm GD (shape\nk ≥ 1\n), or transformation method when\n0 < k < 1\n. Also see Cheng and Feast Algorithm GKM 3 or Marsaglia's squeeze method.\nThe following is a version of the Ahrens-Dieter [acceptance–rejection method](https://bliptext.com/articles/rejection-sampling):\nΓ(δ, 1)\n.\nA summary of this is\nwhere is the integer part of k, ξ is generated via the algorithm above with\nδ =\n(the fractional part of k) and the\nUk\nare all independent.\nWhile the above approach is technically correct, Devroye notes that it is linear in the value of k and generally is not a good choice.\nInstead, he recommends using either rejection-based or table-based methods, depending on context.\nFor example, Marsaglia's simple transformation-rejection method relying on one normal variate X and one uniform variate U:\nWith generates a gamma distributed random number in time that is approximately constant with k. The acceptance rate does depend on k, with an acceptance rate of 0.95, 0.98, and 0.99 for k=1, 2, and 4.\nFor\nk < 1\n, one can use to boost k to be usable with this method.\nIn [Matlab](https://bliptext.com/articles/matlab) numbers can be generated using the function gamrnd, which uses the k, θ representation.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:41:11.384Z","updated_at":"2025-01-25T18:41:11.384Z","search_vector":"'distribut':2 'gamma':1","search_content":" \n**Name:** Gamma \n**Type:** density \n**Pdf Image:** [[Image:Gamma distribution pdf.svg|325px|Probability density plots of gamma distributions]] \n**Cdf Image:** [[Image:Gamma distribution cdf.svg|325px|Cumulative distribution plots of gamma distributions]] \n**Parameters:** k > 0 · [shape](https://bliptext.com/articles/shape-parameter) · θ > 0 · [scale](https://bliptext.com/articles/scale-parameter) \n**Mean:** k \\theta \n**Median:** No simple closed form \n**Mode:** , \n**Variance:** k \\theta^2 \n**Kurtosis:** \\frac{6}{k} \n**Parameters 2:** • α > 0 · shape · • β > 0 · rate \n**Median 2:** No simple closed form \n \n\n# \nIn [probability theory](https://bliptext.com/articles/probability-theory) and [statistics](https://bliptext.com/articles/statistics), the **gamma distribution** is a versatile two-[parameter](https://bliptext.com/articles/statistical-parameter) family of continuous [probability distributions](https://bliptext.com/articles/probability-distribution).\nThe [exponential distribution](https://bliptext.com/articles/exponential-distribution), [Erlang distribution](https://bliptext.com/articles/erlang-distribution), and [chi-squared distribution](https://bliptext.com/articles/chi-squared-distribution) are special cases of the gamma distribution.\nThere are two equivalent parameterizations in common use:\nIn each of these forms, both parameters are positive real numbers.\nThe distribution has important applications in various fields, including [econometrics](https://bliptext.com/articles/econometrics), [Bayesian statistics](https://bliptext.com/articles/bayesian-statistics), life testing.\nIn econometrics, the (k, θ) parameterization is common for modeling waiting times, such as the time until death, where it often takes the form of an [Erlang distribution](https://bliptext.com/articles/erlang-distribution) for integer k values.\nBayesian statistics prefer the (α, β) parameterization, utilizing the gamma distribution as a [conjugate prior](https://bliptext.com/articles/conjugate-prior) for several inverse scale parameters, facilitating analytical tractability in posterior distribution computations.\nThe 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.\nThe 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.\nIts mathematical properties, such as mean, variance, skewness, and higher moments, provide a toolset for statistical analysis and inference.\nPractical applications of the distribution span several disciplines, underscoring its importance in theoretical and applied statistics.\nThe gamma distribution is the [maximum entropy probability distribution](https://bliptext.com/articles/maximum-entropy-probability-distribution) (both with respect to a uniform base measure and a 1/x base measure) for a random variable X for which\nE[X] = kθ = α/β\nis fixed and greater than zero, and\nE[ln X] = ψ(k) + ln θ = ψ(α) − ln β\nis fixed (ψ is the [digamma function](https://bliptext.com/articles/digamma-function)).\n\n# Definitions\nThe parameterization with k and θ appears to be more common in [econometrics](https://bliptext.com/articles/econometrics) and other applied fields, where the gamma distribution is frequently used to model waiting times.\nFor instance, in [life testing](https://bliptext.com/articles/accelerated-life-testing), the waiting time until death is a [random variable](https://bliptext.com/articles/random-variable) that is frequently modeled with a gamma distribution.\nSee Hogg and Craig for an explicit motivation.\nThe parameterization with α and β is more common in [Bayesian statistics](https://bliptext.com/articles/bayesian-statistics), where the gamma distribution is used as a [conjugate prior](https://bliptext.com/articles/conjugate-prior) distribution for various types of inverse scale (rate) parameters, such as the λ of an [exponential distribution](https://bliptext.com/articles/exponential-distribution) or a [Poisson distribution](https://bliptext.com/articles/poisson-distribution) – or for that matter, the β of the gamma distribution itself.\nThe closely related [inverse-gamma distribution](https://bliptext.com/articles/inverse-gamma-distribution) is used as a conjugate prior for scale parameters, such as the [variance](https://bliptext.com/articles/variance) of a [normal distribution](https://bliptext.com/articles/normal-distribution).\nIf k is a positive [integer](https://bliptext.com/articles/integer), then the distribution represents an [Erlang distribution](https://bliptext.com/articles/erlang-distribution); i.e., the sum of k independent [exponentially distributed](https://bliptext.com/articles/exponentially-distributed) [random variables](https://bliptext.com/articles/random-variable), each of which has a mean of θ.\n\n## Characterization using shape α and rate β\nThe gamma distribution can be parameterized in terms of a [shape parameter](https://bliptext.com/articles/shape-parameter)\nα = k\nand an inverse scale parameter\nβ = 1/θ\n, called a [rate parameter](https://bliptext.com/articles/rate-parameter).\nA random variable X that is gamma-distributed with shape α and rate β is denoted\nThe corresponding probability density function in the shape-rate parameterization is\nwhere is the [gamma function](https://bliptext.com/articles/gamma-function).\nFor all positive integers,.\nThe [cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function) is the regularized gamma function:\nwhere is the lower [incomplete gamma function](https://bliptext.com/articles/incomplete-gamma-function).\nIf α is a positive [integer](https://bliptext.com/articles/integer) (i.e., the distribution is an [Erlang distribution](https://bliptext.com/articles/erlang-distribution)), the cumulative distribution function has the following series expansion:\n\n## Characterization using shape k and scale θ\nA random variable X that is gamma-distributed with shape k and scale θ is denoted by\n[[Image:Gamma-PDF-3D.png|thumb|right|320px|Illustration of the gamma PDF for parameter values over k and x with θ set to\n1, 2, 3, 4, 5,\nand\n6\n. One can see each θ layer by itself here as well as by k and x. .]]\nThe [probability density function](https://bliptext.com/articles/probability-density-function) using the shape-scale parametrization is\nHere\nΓ(k)\nis the [gamma function](https://bliptext.com/articles/gamma-function) evaluated at k.\nThe [cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function) is the regularized gamma function:\nwhere is the lower [incomplete gamma function](https://bliptext.com/articles/incomplete-gamma-function).\nIt can also be expressed as follows, if k is a positive [integer](https://bliptext.com/articles/integer) (i.e., the distribution is an [Erlang distribution](https://bliptext.com/articles/erlang-distribution)):\nBoth parametrizations are common because either can be more convenient depending on the situation.\n# [gamma distribution | upload.wikimedia.org/wikipedia/commons/7/78/Gamma-PDF-3D-by-k.png]\n# [gamma distribution | upload.wikimedia.org/wikipedia/commons/9/91/Gamma-PDF-3D-by-Theta.png]\n# [gamma distribution | upload.wikimedia.org/wikipedia/commons/c/cb/Gamma-PDF-3D-by-x.png]\n\n# Properties\n\n## Mean and variance\nThe mean of gamma distribution is given by the product of its shape and scale parameters:\nThe variance is:\nThe square root of the inverse shape parameter gives the [coefficient of variation](https://bliptext.com/articles/coefficient-of-variation):\n\n## Skewness\nThe [skewness](https://bliptext.com/articles/skewness) of the gamma distribution only depends on its shape parameter, k, and it is equal to 2/\\sqrt{k}.\n\n## Higher moments\nThe n-th [raw moment](https://bliptext.com/articles/raw-moment) is given by:\n\n## Median approximations and bounds\nUnlike the mode and the mean, which have readily calculable formulas based on the parameters, the median does not have a closed-form equation.\nThe median for this distribution is the value \\nu such that\nA 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 \\theta = 1)\nwhere \\mu(k) = k is the mean and \\nu(k) is the median of the distribution.\nFor other values of the scale parameter, the mean scales to, and the median bounds and approximations would be similarly scaled by θ.\nK. P. Choi found the first five terms in a [Laurent series](https://bliptext.com/articles/laurent-series) asymptotic approximation of the median by comparing the median to [Ramanujan's \\theta function](https://bliptext.com/articles/ramanujan-theta-function).\nBerg and Pedersen found more terms:\nPartial sums of these series are good approximations for high enough k; they are not plotted in the figure, which is focused on the low-k region that is less well approximated.\nBerg and Pedersen also proved many properties of the median, showing that it is a convex function of k, and that the asymptotic behavior near k = 0 is (where γ is the [Euler–Mascheroni constant](https://bliptext.com/articles/euler-mascheroni-constant)), and that for all k > 0 the median is bounded by.\nA closer linear upper bound, for k \\ge 1 only, was provided in 2021 by Gaunt and Merkle, relying on the Berg and Pedersen result that the slope of \\nu(k) is everywhere less than 1:\nfor k \\ge 1 (with equality at k = 1)\nwhich can be extended to a bound for all k > 0 by taking the max with the chord shown in the figure, since the median was proved convex.\nAn approximation to the median that is asymptotically accurate at high k and reasonable down to k = 0.5 or a bit lower follows from the [Wilson–Hilferty transformation](https://bliptext.com/articles/wilson-hilferty-transformation):\nwhich goes negative for k 0.\nIn particular, he proposed these closed-form bounds, which he proved in 2023:\nis a lower bound, asymptotically tight as is an upper bound, asymptotically tight as k \\to 0\nLyon also showed (informally in 2021, rigorously in 2023) two other lower bounds that are not [closed-form expressions](https://bliptext.com/articles/closed-form-expression), including this one involving the [gamma function](https://bliptext.com/articles/gamma-function), based on solving the integral expression substituting 1 for e^{-x}:\n(approaching equality as k \\to 0)\nand the tangent line at k = 1 where the derivative was found to be :\n(with equality at k = 1)\nwhere Ei is the [exponential integral](https://bliptext.com/articles/exponential-integral).\nAdditionally, he showed that interpolations between bounds could provide excellent approximations or tighter bounds to the median, including an approximation that is exact at k = 1 (where ) and has a maximum relative error less than 0.6%.\nInterpolated approximations and bounds are all of the form\nwhere \\tilde{g} is an interpolating function running monotonically from 0 at low k to 1 at high k, approximating an ideal, or exact, interpolator g(k):\nFor the simplest interpolating function considered, a first-order rational function\nthe tightest lower bound has\nand the tightest upper bound has\nThe interpolated bounds are plotted (mostly inside the yellow region) in the [log–log plot](https://bliptext.com/articles/log-log-plot) shown.\nEven tighter bounds are available using different interpolating functions, but not usually with closed-form parameters like these.\n# [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 before 2021. | upload.wikimedia.org/wikipedia/commons/4/42/Gamma///distribution///median///bounds.png]\n# [Two gamma distribution median asymptotes which were proved in 2023 to be bounds (upper solid red and lower dashed red), of the from, 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. | upload.wikimedia.org/wikipedia/commons/1/1e/Gamma///distribution///median///Lyon///bounds.png]\n# [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 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. | upload.wikimedia.org/wikipedia/commons/a/a7/Gamma///distribution///median///loglog///bounds.png]\n\n## Summation\nIf\nXi\nhas a\nGamma(ki, θ)\ndistribution for\ni = 1, 2, ..., N\n(i.e., all distributions have the same scale parameter θ), then\nprovided all\nXi\nare [independent](https://bliptext.com/articles/statistical-independence).\nFor the cases where the\nXi\nare [independent](https://bliptext.com/articles/statistical-independence) but have different scale parameters, see Mathai or Moschopoulos.\nThe gamma distribution exhibits [infinite divisibility](https://bliptext.com/articles/infinite-divisibility-probability).\n\n## Scaling\nIf\nthen, for any\nc > 0\n,\nby moment generating functions,\nor equivalently, if\n(shape-rate parameterization)\nIndeed, we know that if X is an [exponential r.v.](https://bliptext.com/articles/exponential-distribution) with rate λ, then\ncX\nis an exponential r.v. with rate\nλ/c\n\n## Exponential family\nThe gamma distribution is a two-parameter [exponential family](https://bliptext.com/articles/exponential-family) with [natural parameters](https://bliptext.com/articles/natural-parameters)\nk − 1\nand\n−1/θ\n(equivalently,\nα − 1\nand\n−β\n), and [natural statistics](https://bliptext.com/articles/natural-statistics) X and\nln X\n.\nIf the shape parameter k is held fixed, the resulting one-parameter family of distributions is a [natural exponential family](https://bliptext.com/articles/natural-exponential-family).\n\n## Logarithmic expectation and variance\nOne can show that\nor equivalently,\nwhere ψ is the [digamma function](https://bliptext.com/articles/digamma-function).\nLikewise,\nwhere \\psi^{(1)} is the [trigamma function](https://bliptext.com/articles/trigamma-function).\nThis can be derived using the [exponential family](https://bliptext.com/articles/exponential-family) formula for the [moment generating function of the sufficient statistic](https://bliptext.com/articles/exponential-family), because one of the sufficient statistics of the gamma distribution is\nln x\n.\n\n## Information entropy\nThe [information entropy](https://bliptext.com/articles/information-entropy) is\nIn the k, θ parameterization, the [information entropy](https://bliptext.com/articles/information-entropy) is given by\n\n## Kullback–Leibler divergence\n[[Image:Gamma-KL-3D.png|thumb|right|320px|Illustration of the Kullback–Leibler (KL) divergence for two gamma PDFs. Here\nβ = β0 + 1\nwhich are set to\n1, 2, 3, 4, 5,\nand\n6\n. The typical asymmetry for the KL divergence is clearly visible.]]\nThe [Kullback–Leibler divergence](https://bliptext.com/articles/kullback-leibler-divergence) (KL-divergence), of\nGamma(αp, βp)\n(\"true\" distribution) from\nGamma(αq, βq)\n(\"approximating\" distribution) is given by\nWritten using the k, θ parameterization, the KL-divergence of\nGamma(kp, θp)\nfrom\nGamma(kq, θq)\nis given by\n\n## Laplace transform\nThe [Laplace transform](https://bliptext.com/articles/laplace-transform) of the gamma PDF is\n\n# Related distributions\n\n## General\nX ~ Gamma(1, λ)\n(in the shape–rate parametrization), then X has an [exponential distribution](https://bliptext.com/articles/exponential-distribution) with rate parameter λ.\nIn the shape-scale parametrization,\nX ~ Gamma(1, λ)\nhas an exponential distribution with rate parameter\n1/λ\n.\nX ~ Gamma(ν/2, 2)\n(in the shape–scale parametrization), then X is identical to\nχ2(ν)\n, the [chi-squared distribution](https://bliptext.com/articles/chi-squared-distribution) with ν degrees of freedom.\nConversely, if\nQ ~ χ2(ν)\nand c is a positive constant, then\ncQ ~ Gamma(ν/2, 2c)\n.\nθ = 1/k\n, one obtains the [Schulz-Zimm distribution](https://bliptext.com/articles/schulz-zimm-distribution), which is most prominently used to model polymer chain lengths.\n1/θ\n. If\nX ~ Gamma(k, θ)\n, then \\log X follows an exponential-gamma (abbreviated exp-gamma) distribution.\nIt is sometimes referred to as the log-gamma distribution.\n[](https://bliptext.com/articles/)F[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)r[](https://bliptext.com/articles/)m[](https://bliptext.com/articles/)u[](https://bliptext.com/a rticles/)l[](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)s[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)f[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)r[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)i[](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)s[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)m[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/)d[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)v[](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)r[](https://bliptext.com/articles/)i[](https://bliptext.com/articles/)a[](https://bliptext.com/a rticles/)n[](https://bliptext.com/articles/)c[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)r[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)i[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)h[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)s[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/)c[](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)i[](https://bliptext.com/a rticles/)o[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/).[](https://bliptext.com/articles/)\nX ~ Gamma(k, θ)\n, then \\sqrt{X} follows a [generalized gamma distribution](https://bliptext.com/articles/generalized-gamma-distribution) with parameters\np = 2\n,\nd = 2k\n, and.\nX ~ Gamma(k,θ)\n, then X^q for q > 0 follows a [generalized gamma distribution](https://bliptext.com/articles/generalized-gamma-distribution) with parameters\np = 1/q\n,\nd = k/q\n, and.\nX ~ Gamma(k, θ)\nwith shape k and scale θ, then\n1/X ~ Inv-Gamma(k, θ−1)\n(see [Inverse-gamma distribution](https://bliptext.com/articles/inverse-gamma-distribution) for derivation).\nX ~ Gamma(α, θ)\nand\nY ~ Gamma(β, θ)\nare independently distributed, then\nX/(X + Y)\nhas a [beta distribution](https://bliptext.com/articles/beta-distribution) with parameters α and β, and\nX/(X + Y)\nis independent of\nX + Y\n, which is\nGamma(α + β, θ)\n-distributed.\nXi ~ Gamma(αi, 1)\nare independently distributed, then the vector (\nX1/S, ..., Xn/S)\n, where\nS = X1 + ...\n+ Xn\n, follows a [Dirichlet distribution](https://bliptext.com/articles/dirichlet-distribution) with parameters\nα1, ..., αn\n.\nμ = kθ\nand variance\nσ2 = kθ2\n.\n\n## Compound gamma\nIf 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.\nThe [compound distribution](https://bliptext.com/articles/compound-distribution), which results from integrating out the inverse scale, has a closed-form solution known as the [compound gamma distribution](https://bliptext.com/articles/compound-gamma-distribution).\nIf, 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](https://bliptext.com/articles/k-distribution).\n\n## Weibull and stable count\nThe gamma distribution can be expressed as the product distribution of a [Weibull distribution](https://bliptext.com/articles/weibull-distribution) and a variant form of the [stable count distribution](https://bliptext.com/articles/stable-count-distribution).\nIts shape parameter k can be regarded as the inverse of Lévy's stability parameter in the stable count distribution:\nwhere is a standard stable count distribution of shape, and W_k(x) is a standard Weibull distribution of shape k.\n\n# Statistical inference\n\n## Parameter estimation\n\n### Maximum likelihood estimation\nThe likelihood function for N [iid](https://bliptext.com/articles/iid) observations\n(x1, ..., xN)\nis\nfrom which we calculate the log-likelihood function\nFinding the maximum with respect to θ by taking the derivative and setting it equal to zero yields the [maximum likelihood](https://bliptext.com/articles/maximum-likelihood) estimator of the θ parameter, which equals the [sample mean](https://bliptext.com/articles/sample-mean) \\bar{x} divided by the shape parameter k:\nSubstituting this into the log-likelihood function gives\nWe need at least two samples: N\\ge2, because for N=1, the function \\ell(k) increases without bounds as k\\to\\infty.\nFor k>0, it can be verified that \\ell(k) is strictly [concave](https://bliptext.com/articles/concave-function), by using [inequality properties of the polygamma function](https://bliptext.com/articles/polygamma-function).\nFinding the maximum with respect to k by taking the derivative and setting it equal to zero yields\nwhere ψ is the [digamma function](https://bliptext.com/articles/digamma-function) and is the sample mean of\nln x\n. There is no closed-form solution for k. The function is numerically very well behaved, so if a numerical solution is desired, it can be found using, for example, [Newton's method](https://bliptext.com/articles/newton-s-method).\nAn initial value of k can be found either using the [method of moments](https://bliptext.com/articles/method-of-moments-statistics), or using the approximation\nIf we let\nthen k is approximately\nwhich is within 1.5% of the correct value.\nAn explicit form for the Newton–Raphson update of this initial guess is:\nAt the maximum-likelihood estimate, the expected values for x and \\ln x agree with the empirical averages:\n\n#### Caveat for small shape parameter\nFor data,, that is represented in a [floating point](https://bliptext.com/articles/floating-point) format that underflows to 0 for values smaller than \\varepsilon, the logarithms that are needed for the maximum-likelihood estimate will cause failure if there are any underflows.\nIf we assume the data was generated by a gamma distribution with cdf, then the probability that there is at least one underflow is:\nThis probability will approach 1 for small k and large N. For example, at k=10^{-2}, N=10^4 and,.\nA workaround is to instead have the data in logarithmic format.\nIn 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 k<1.\nFollowing the implementation in, this can be done as follows: sample and independently.\nThen the required logarithmic sample is, so that.\n\n### Closed-form estimators\nThere exist consistent closed-form estimators of k and θ that are derived from the likelihood of the [generalized gamma distribution](https://bliptext.com/articles/generalized-gamma-distribution).\nThe estimate for the shape k is\nand the estimate for the scale θ is\nUsing the sample mean of x, the sample mean of\nln x\n, and the sample mean of the product\nx·ln x\nsimplifies the expressions to:\nIf the rate parameterization is used, the estimate of.\nThese estimators are not strictly maximum likelihood estimators, but are instead referred to as mixed type log-moment estimators.\nThey have however similar efficiency as the maximum likelihood estimators.\nAlthough these estimators are consistent, they have a small bias.\nA bias-corrected variant of the estimator for the scale θ is\nA bias correction for the shape parameter k is given as\n\n### Bayesian minimum mean squared error\nWith known k and unknown θ, the posterior density function for theta (using the standard scale-invariant [prior](https://bliptext.com/articles/prior-probability) for θ) is\nDenoting\nIntegration with respect to θ can be carried out using a change of variables, revealing that\n1/θ\nis gamma-distributed with parameters\nα = Nk\n,\nβ = y\n.\nThe moments can be computed by taking the ratio (m by\nm = 0\n)\nwhich shows that the mean ± standard deviation estimate of the posterior distribution for θ is\n\n## Bayesian inference\n\n### Conjugate prior\nIn [Bayesian inference](https://bliptext.com/articles/bayesian-inference), the **gamma distribution** is the [conjugate prior](https://bliptext.com/articles/conjugate-prior) to many likelihood distributions: the [Poisson](https://bliptext.com/articles/poisson-distribution), [exponential](https://bliptext.com/articles/exponential-distribution), [normal](https://bliptext.com/articles/normal-distribution) (with known mean), [Pareto](https://bliptext.com/articles/pareto-distribution), gamma with known shape σ, [inverse gamma](https://bliptext.com/articles/inverse-gamma) with known shape parameter, and [Gompertz](https://bliptext.com/articles/gompertz-distribution) with known scale parameter.\nThe gamma distribution's [conjugate prior](https://bliptext.com/articles/conjugate-prior) is:\nwhere Z is the normalizing constant with no closed-form solution.\nThe posterior distribution can be found by updating the parameters as follows:\nwhere n is the number of observations, and\nxi\nis the i-th observation.\n\n# Occurrence and applications\nConsider a sequence of events, with the waiting time for each event being an exponential distribution with rate β.\nThen the waiting time for the n-th event to occur is the gamma distribution with integer shape \\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.\nExamples include the waiting time of [cell-division events](https://bliptext.com/articles/cell-division), number of compensatory mutations for a given mutation, waiting time until a repair is necessary for a hydraulic system, and so on.\nIn biophysics, the dwell time between steps of a molecular motor like [ATP synthase](https://bliptext.com/articles/atp-synthase) is nearly exponential at constant ATP concentration, revealing that each step of the motor takes a single ATP hydrolysis.\nIf there were n ATP hydrolysis events, then it would be a gamma distribution with degree n.\nThe gamma distribution has been used to model the size of [insurance claims](https://bliptext.com/articles/insurance-policy) and rainfalls.\nThis means that aggregate insurance claims and the amount of rainfall accumulated in a reservoir are modelled by a [gamma process](https://bliptext.com/articles/gamma-process) – much like the [exponential distribution](https://bliptext.com/articles/exponential-distribution) generates a [Poisson process](https://bliptext.com/articles/poisson-process).\nThe gamma distribution is also used to model errors in multi-level [Poisson regression](https://bliptext.com/articles/poisson-regression) models because a [mixture](https://bliptext.com/articles/mixture-distribution) of [Poisson distributions](https://bliptext.com/articles/poisson-distribution) with gamma-distributed rates has a known closed form distribution, called [negative binomial](https://bliptext.com/articles/negative-binomial).\nIn wireless communication, the gamma distribution is used to model the [multi-path fading](https://bliptext.com/articles/multi-path-fading) of signal power; see also [Rayleigh distribution](https://bliptext.com/articles/rayleigh-distribution) and [Rician distribution](https://bliptext.com/articles/rician-distribution).\nIn [oncology](https://bliptext.com/articles/oncology), the age distribution of [cancer](https://bliptext.com/articles/cancer) [incidence](https://bliptext.com/articles/disease-incidence) often follows the gamma distribution, wherein the shape and scale parameters predict, respectively, the number of [driver events](https://bliptext.com/articles/carcinogenesis) and the time interval between them.\nIn [neuroscience](https://bliptext.com/articles/neuroscience), the gamma distribution is often used to describe the distribution of [inter-spike intervals](https://bliptext.com/articles/temporal-coding).\nIn [bacterial](https://bliptext.com/articles/bacterial-genetics) [gene expression](https://bliptext.com/articles/gene-expression), the [copy number](https://bliptext.com/articles/copy-number-analysis) of a [constitutively expressed](https://bliptext.com/articles/constitutively-expressed) protein often follows the gamma distribution, where the scale and shape parameter are, respectively, the mean number of bursts per cell cycle and the mean number of [protein molecules](https://bliptext.com/articles/protein-molecule) produced by a single mRNA during its lifetime.\nIn [genomics](https://bliptext.com/articles/genomics), the gamma distribution was applied in [peak calling](https://bliptext.com/articles/peak-calling) step (i.e., in recognition of signal) in [ChIP-chip](https://bliptext.com/articles/chip-chip) and [ChIP-seq](https://bliptext.com/articles/chip-seq) data analysis.\nIn Bayesian statistics, the gamma distribution is widely used as a [conjugate prior](https://bliptext.com/articles/conjugate-prior).\nIt is the conjugate prior for the [precision](https://bliptext.com/articles/precision-statistics) (i.e. inverse of the variance) of a [normal distribution](https://bliptext.com/articles/normal-distribution).\nIt is also the conjugate prior for the [exponential distribution](https://bliptext.com/articles/exponential-distribution).\nIn [phylogenetics](https://bliptext.com/articles/phylogenetics), the gamma distribution is the most commonly used approach to model among-sites rate variation when [maximum likelihood](https://bliptext.com/articles/computational-phylogenetics), [Bayesian](https://bliptext.com/articles/bayesian-inference-in-phylogeny), or [distance matrix methods](https://bliptext.com/articles/distance-matrices-in-phylogeny) are used to estimate phylogenetic trees.\nPhylogenetic analyzes that use the gamma distribution to model rate variation estimate a single parameter from the data because they limit consideration to distributions where\nα = β\n. This parameterization means that the mean of this distribution is 1 and the variance is\n1/α\n. Maximum likelihood and Bayesian methods typically use a discrete approximation to the continuous gamma distribution.\n\n# Random variate generation\nGiven the scaling property above, it is enough to generate gamma variables with\nθ = 1\n, as we can later convert to any value of β with a simple division.\nSuppose we wish to generate random variables from\nGamma(n + δ, 1)\n, where n is a non-negative integer and\n0 < δ < 1\n. Using the fact that a\nGamma(1, 1)\ndistribution is the same as an\nExp(1)\ndistribution, and noting the method of [generating exponential variables](https://bliptext.com/articles/exponential-distribution), we conclude that if U is [uniformly distributed](https://bliptext.com/articles/uniform-distribution-continuous) on (0, 1], then\n−ln U\nis distributed\nGamma(1, 1)\n(i.e. [inverse transform sampling](https://bliptext.com/articles/inverse-transform-sampling)).\nNow, using the \"α-addition\" property of gamma distribution, we expand this result:\nwhere\nUk\nare all uniformly distributed on (0, 1] and [independent](https://bliptext.com/articles/statistical-independence). All that is left now is to generate a variable distributed as\nGamma(δ, 1)\nfor\n0 < δ < 1\nand apply the \"α-addition\" property once more.\nThis is the most difficult part.\nRandom generation of gamma variates is discussed in detail by Devroye, noting that none are uniformly fast for all shape parameters.\nFor small values of the shape parameter, the algorithms are often not valid.\nFor arbitrary values of the shape parameter, one can apply the Ahrens and Dieter modified acceptance-rejection method Algorithm GD (shape\nk ≥ 1\n), or transformation method when\n0 < k < 1\n. Also see Cheng and Feast Algorithm GKM 3 or Marsaglia's squeeze method.\nThe following is a version of the Ahrens-Dieter [acceptance–rejection method](https://bliptext.com/articles/rejection-sampling):\nΓ(δ, 1)\n.\nA summary of this is\nwhere is the integer part of k, ξ is generated via the algorithm above with\nδ =\n(the fractional part of k) and the\nUk\nare all independent.\nWhile the above approach is technically correct, Devroye notes that it is linear in the value of k and generally is not a good choice.\nInstead, he recommends using either rejection-based or table-based methods, depending on context.\nFor example, Marsaglia's simple transformation-rejection method relying on one normal variate X and one uniform variate U:\nWith generates a gamma distributed random number in time that is approximately constant with k. The acceptance rate does depend on k, with an acceptance rate of 0.95, 0.98, and 0.99 for k=1, 2, and 4.\nFor\nk < 1\n, one can use to boost k to be usable with this method.\nIn [Matlab](https://bliptext.com/articles/matlab) numbers can be generated using the function gamrnd, which uses the k, θ representation.","revisionCount":0},"hasMore":false,"nextCursor":null}

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