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In probability theory and statistics, the **gamma distribution** is a versatile two-parameter family of continuous probability distributions.^{[1]} The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution.^{[2]} There are two equivalent parameterizations in common use:

- With a shape parameter k and a scale parameter θ
- With a shape parameter and an inverse scale parameter , called a rate parameter.

Probability density function | |||

Cumulative distribution function | |||

Parameters | |||
---|---|---|---|

Support | |||

CDF | |||

Mean | |||

Median | No simple closed form | No simple closed form | |

Mode | , | ||

Variance | |||

Skewness | |||

Excess kurtosis | |||

Entropy | |||

MGF | |||

CF | |||

Fisher information | |||

Method of moments |

In each of these forms, both parameters are positive real numbers.

The distribution has important applications in various fields, including econometrics, Bayesian statistics, life testing.^{[3]} In 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 for integer k values. Bayesian statistics prefer the (α, β) parameterization, utilizing the gamma distribution as a conjugate prior for several inverse scale parameters, facilitating analytical tractability in posterior distribution computations. The probability density and cumulative distribution functions of the gamma distribution vary based on the chosen parameterization, both offering insights into the behavior of gamma-distributed random variables. The gamma distribution is integral to modeling a range of phenomena due to its flexible shape, which can capture various statistical distributions, including the exponential and chi-squared distributions under specific conditions. Its mathematical properties, such as mean, variance, skewness, and higher moments, provide a toolset for statistical analysis and inference. Practical applications of the distribution span several disciplines, underscoring its importance in theoretical and applied statistics.^{[4]}

The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a base measure) for a random variable X for which **E**[*X*] = *kθ* = *α*/*β* is fixed and greater than zero, and **E**[ln *X*] = *ψ*(*k*) + ln *θ* = *ψ*(*α*) − ln *β* is fixed (ψ is the digamma function).^{[5]}

The parameterization with k and 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^{[6]} for an explicit motivation.

The parameterization with α and β 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 λ of an exponential distribution or a Poisson distribution^{[7]} – or for that matter, the β 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 k is a positive integer, then the distribution represents an Erlang distribution; i.e., the sum of k independent exponentially distributed random variables, each of which has a mean of θ.

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

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

where is the gamma function. For all positive integers, .

The cumulative distribution function is the regularized gamma function:

where 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:^{[8]}

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

The probability density function using the shape-scale parametrization is

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

The cumulative distribution function is the regularized gamma function:

where 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):^{[8]}

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:

The variance is:

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

The skewness of the gamma distribution only depends on its shape parameter, k, and it is equal to

The n-th raw moment is given by:

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 ν such that

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 )

where is the mean and is the median of the distribution.^{[9]} For other values of the scale parameter, the mean scales to , and the median bounds and approximations would be similarly scaled by 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 function.^{[10]} Berg and Pedersen found more terms:^{[11]}

Partial 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.

Berg and Pedersen also proved many properties of the median, showing that it is a convex function of k,^{[12]} and that the asymptotic behavior near is (where γ is the Euler–Mascheroni constant), and that for all the median is bounded by .^{[11]}

A closer linear upper bound, for only, was provided in 2021 by Gaunt and Merkle,^{[13]} relying on the Berg and Pedersen result that the slope of is everywhere less than 1:

- for (with equality at )

which can be extended to a bound for all by taking the max with the chord shown in the figure, since the median was proved convex.^{[12]}

An approximation to the median that is asymptotically accurate at high k and reasonable down to or a bit lower follows from the Wilson–Hilferty transformation:

which goes negative for .

In 2021, Lyon proposed several approximations of the form . He conjectured values of A and B for which this approximation is an asymptotically tight upper or lower bound for all .^{[14]} In particular, he proposed these closed-form bounds, which he proved in 2023:^{[15]}

- is a lower bound, asymptotically tight as
- is an upper bound, asymptotically tight as

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 :

- (approaching equality as )

and the tangent line at where the derivative was found to be :

- (with equality at )

where Ei is the exponential integral.^{[14]}^{[15]}

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

where is an interpolating function running monotonically from 0 at low k to 1 at high k, approximating an ideal, or exact, interpolator :

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

the tightest lower bound has

and the tightest upper bound has

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.^{[14]}

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

provided all *X*_{i} are independent.

For the cases where the *X*_{i} are independent but have different scale parameters, see Mathai ^{[16]} or Moschopoulos.^{[17]}

The gamma distribution exhibits infinite divisibility.

If

then, for any *c* > 0,

- by moment generating functions,

or equivalently, if

- (shape-rate parameterization)

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).

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.

One can show that

or equivalently,

where ψ is the digamma function. Likewise,

where 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*.

The information entropy is

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

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^{[18]}

Written using the k, θ parameterization, the KL-divergence of Gamma(*k*_{p}, *θ*_{p}) from Gamma(*k*_{q}, *θ*_{q}) is given by

The Laplace transform of the gamma PDF is

- Let be independent and identically distributed random variables following an exponential distribution with rate parameter λ, then ~ Gamma(n, λ) where n is the shape parameter and λ is the rate, and .
- 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*χ*^{2}(*ν*), the chi-squared distribution with ν degrees of freedom. Conversely, if*Q*~*χ*^{2}(*ν*) and c is a positive constant, then*cQ*~ Gamma(*ν*/2, 2*c*). - 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 k-th "arrival" in a one-dimensional Poisson process with intensity 1/
*θ*. If

- then

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

- If
*X*~ Gamma(*k*,*θ*), then follows an exponential-gamma (abbreviated exp-gamma) distribution.^{[19]}It is sometimes referred to as the log-gamma distribution.^{[20]}Formulas for its mean and variance are in the section #Logarithmic expectation and variance. - If
*X*~ Gamma(*k*,*θ*), then follows a generalized gamma distribution with parameters*p*= 2,*d*= 2*k*, and^{[citation needed]}. - More generally, if
*X*~ Gamma(*k*,*θ*), then for follows a generalized gamma distribution with parameters*p*= 1/*q*,*d*=*k*/*q*, and . - 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 are independent, then , or equivalently,
- Parametrization 2: If are independent, then , or equivalently,
- 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 (*X*_{1}/*S*, ...,*X*_{n}/*S*), where*S*=*X*_{1}+ ... +*X*_{n}, follows a Dirichlet distribution with parameters*α*_{1}, ...,*α*_{n}. - For large k the gamma distribution converges to normal distribution with mean
*μ*=*kθ*and variance*σ*^{2}=*kθ*^{2}. - 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.
^{[21]}The corresponding density is , where denotes the Fox–Wright Psi function. - For the shape-scale parameterization , if the scale parameter where denotes the Inverse-gamma distribution, then the marginal distribution where denotes the Beta prime distribution.

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.^{[22]}

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 can be expressed as the product distribution of a Weibull distribution and a variant form of the stable count distribution. Its shape parameter can be regarded as the inverse of Lévy's stability parameter in the stable count distribution: where is a standard stable count distribution of shape , and is a standard Weibull distribution of shape .

The likelihood function for N iid observations (*x*_{1}, ..., *x*_{N}) is

from which we calculate the log-likelihood function

Finding the maximum with respect to θ by taking the derivative and setting it equal to zero yields the maximum likelihood estimator of the θ parameter, which equals the sample mean divided by the shape parameter k:

Substituting this into the log-likelihood function gives

We need at least two samples: , because for , the function increases without bounds as . For , it can be verified that is strictly concave, by using inequality properties of the polygamma function. Finding the maximum with respect to k by taking the derivative and setting it equal to zero yields

where ψ is the digamma function and is the sample mean of ln *x*. 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. An initial value of k can be found either using the method of moments, or using the approximation

If we let

then k is approximately

which is within 1.5% of the correct value.^{[23]} An explicit form for the Newton–Raphson update of this initial guess is:^{[24]}

At the maximum-likelihood estimate , the expected values for x and agree with the empirical averages:

For data, , that is represented in a floating point format that underflows to 0 for values smaller than , 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 , then the probability that there is at least one underflow is:

This probability will approach 1 for small k and large N. For example, at , and , . 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:^{[25]} sample and independently. Then the required logarithmic sample is , so that .

There exist consistent closed-form estimators of k and θ that are derived from the likelihood of the generalized gamma distribution.^{[26]}

The estimate for the shape k is

and the estimate for the scale θ is

Using the sample mean of x, the sample mean of ln *x*, and the sample mean of the product *x*·ln *x* simplifies the expressions to:

If the rate parameterization is used, the estimate of .

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 θ is

A bias correction for the shape parameter k is given as^{[27]}

With known k and unknown θ, the posterior density function for theta (using the standard scale-invariant prior for θ) is

Denoting

Integration with respect to θ can be carried out using a change of variables, revealing that 1/*θ* is gamma-distributed with parameters *α* = *Nk*, *β* = *y*.

The moments can be computed by taking the ratio (m by *m* = 0)

which shows that the mean ± standard deviation estimate of the posterior distribution for θ is

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 σ, inverse gamma with known shape parameter, and Gompertz with known scale parameter.

The gamma distribution's conjugate prior is:^{[28]}

where Z is the normalizing constant with no closed-form solution. The posterior distribution can be found by updating the parameters as follows:

where n is the number of observations, and *x*_{i} is the i-th observation.

Consider a sequence of events, with the waiting time for each event being an exponential distribution with rate β. Then the waiting time for the n-th event to occur is the gamma distribution with integer shape . 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.^{[29]} Examples include the waiting time of cell-division events,^{[30]} number of compensatory mutations for a given mutation,^{[31]} waiting time until a repair is necessary for a hydraulic system,^{[32]} 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.^{[33]}

The gamma distribution has been used to model the size of insurance claims^{[34]} and rainfalls.^{[35]} 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]} 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.^{[36]}^{[37]}

In neuroscience, the gamma distribution is often used to describe the distribution of inter-spike intervals.^{[38]}^{[39]}

In bacterial gene expression, the copy number of a 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 produced by a single mRNA during its lifetime.^{[40]}

In genomics, the gamma distribution was applied in peak calling step (i.e., in recognition of signal) in ChIP-chip^{[41]} and ChIP-seq^{[42]} 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^{[43]} 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 *α* = *β*. This parameterization means that the mean of this distribution is 1 and the variance is 1/*α*. Maximum likelihood and Bayesian methods typically use a discrete approximation to the continuous gamma distribution.^{[44]}^{[45]}

Given the scaling property above, it is enough to generate gamma variables with *θ* = 1, as we can later convert to any value of β with a simple division.

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

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

Random generation of gamma variates is discussed in detail by Devroye,^{[46]}^{: 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.^{[46]}^{: 406 } For arbitrary values of the shape parameter, one can apply the Ahrens and Dieter^{[47]} modified acceptance-rejection method Algorithm GD (shape *k* ≥ 1), or transformation method^{[48]} when 0 < *k* < 1. Also see Cheng and Feast Algorithm GKM 3^{[49]} or Marsaglia's squeeze method.^{[50]}

The following is a version of the Ahrens-Dieter acceptance–rejection method:^{[47]}

- Generate U, V and W as iid uniform (0, 1] variates.
- If then and . Otherwise, and .
- If then go to step 1.
- ξ is distributed as Γ(
*δ*, 1).

A summary of this is

where is the integer part of k, ξ is generated via the algorithm above with *δ* = {*k*} (the fractional part of k) and the *U*_{k} are all independent.

While the above approach is technically correct, Devroye notes that it is linear in the value of k and generally is not a good choice. Instead, he recommends using either rejection-based or table-based methods, depending on context.^{[46]}^{: 401–428 }

For example, Marsaglia's simple transformation-rejection method relying on one normal variate X and one uniform variate U:^{[25]}

- Set and .
- Set .
- If and return , else go back to step 2.

With 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. For *k* < 1, one can use to boost k to be usable with this method.

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

**^**"Gamma distribution | Probability, Statistics, Distribution | Britannica".*www.britannica.com*. Archived from the original on 2024-05-19. Retrieved 2024-10-09.**^**Weisstein, Eric W. "Gamma Distribution".*mathworld.wolfram.com*. Archived from the original on 2024-05-28. Retrieved 2024-10-09.**^**"Gamma Distribution | Gamma Function | Properties | PDF".*www.probabilitycourse.com*. Archived from the original on 2024-06-13. Retrieved 2024-10-09.**^**"4.5: Exponential and Gamma Distributions".*Statistics LibreTexts*. 2019-03-11. Retrieved 2024-10-10.**^**Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model" (PDF).*Journal of Econometrics*.**150**(2): 219–230. CiteSeerX 10.1.1.511.9750. doi:10.1016/j.jeconom.2008.12.014. Archived from the original (PDF) on 2016-03-07. Retrieved 2011-06-02.**^**Hogg, R. V.; Craig, A. T. (1978).*Introduction to Mathematical Statistics*(4th ed.). New York: Macmillan. pp. Remark 3.3.1. ISBN 0023557109.**^**Gopalan, Prem; Hofman, Jake M.; Blei, David M. (2013). "Scalable Recommendation with Poisson Factorization". arXiv:1311.1704 [cs.IR].- ^
^{a}^{b}Papoulis, Pillai,*Probability, Random Variables, and Stochastic Processes*, Fourth Edition **^**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, [1] Archived 2024-10-09 at the Wayback Machine.**^**Choi, K. P. "On the Medians of the Gamma Distributions and an Equation of Ramanujan" Archived 2021-01-23 at the Wayback Machine, Proceedings of the American Mathematical Society, Vol. 121, No. 1 (May, 1994), pp. 245–251.- ^
^{a}^{b}Berg, Christian & Pedersen, Henrik L. (March 2006). "The Chen–Rubin conjecture in a continuous setting" (PDF).*Methods and Applications of Analysis*.**13**(1): 63–88. doi:10.4310/MAA.2006.v13.n1.a4. S2CID 6704865. Archived (PDF) from the original on 16 January 2021. Retrieved 1 April 2020. - ^
^{a}^{b}Berg, Christian and Pedersen, Henrik L. "Convexity of the median in the gamma distribution" Archived 2023-05-26 at the Wayback Machine. **^**Gaunt, Robert E., and Milan Merkle (2021). "On bounds for the mode and median of the generalized hyperbolic and related distributions".*Journal of Mathematical Analysis and Applications*.**493**(1): 124508. arXiv:2002.01884. doi:10.1016/j.jmaa.2020.124508. S2CID 221103640.`{{cite journal}}`

: CS1 maint: multiple names: authors list (link)- ^
^{a}^{b}^{c}Lyon, Richard F. (13 May 2021). "On closed-form tight bounds and approximations for the median of a gamma distribution".*PLOS One*.**16**(5): e0251626. arXiv:2011.04060. Bibcode:2021PLoSO..1651626L. doi:10.1371/journal.pone.0251626. PMC 8118309. PMID 33984053. - ^
^{a}^{b}Lyon, Richard F. (13 May 2021). "Tight bounds for the median of a gamma distribution".*PLOS One*.**18**(9): e0288601. doi:10.1371/journal.pone.0288601. PMC 10490949. PMID 37682854. **^**Mathai, A. M. (1982). "Storage capacity of a dam with gamma type inputs".*Annals of the Institute of Statistical Mathematics*.**34**(3): 591–597. doi:10.1007/BF02481056. ISSN 0020-3157. S2CID 122537756.**^**Moschopoulos, P. G. (1985). "The distribution of the sum of independent gamma random variables".*Annals of the Institute of Statistical Mathematics*.**37**(3): 541–544. doi:10.1007/BF02481123. S2CID 120066454.**^**Penny, W. D. "KL-Divergences of Normal, Gamma, Dirichlet, and Wishart densities".**^**"ExpGammaDistribution—Wolfram Language Documentation".**^**"scipy.stats.loggamma — SciPy v1.8.0 Manual".*docs.scipy.org*.**^**Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). "The Modified-Half-Normal distribution: Properties and an efficient sampling scheme" (PDF).*Communications in Statistics - Theory and Methods*.**52**(5): 1591–1613. doi:10.1080/03610926.2021.1934700. ISSN 0361-0926. S2CID 237919587. Retrieved 2 September 2022.**^**Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions".*Metrika*.**16**: 27–31. doi:10.1007/BF02613934. S2CID 123366328.**^**Minka, Thomas P. (2002). "Estimating a Gamma distribution" (PDF).**^**Choi, S. C.; Wette, R. (1969). "Maximum Likelihood Estimation of the Parameters of the Gamma Distribution and Their Bias".*Technometrics*.**11**(4): 683–690. doi:10.1080/00401706.1969.10490731.- ^
^{a}^{b}Marsaglia, G.; Tsang, W. W. (2000). "A simple method for generating gamma variables".*ACM Transactions on Mathematical Software*.**26**(3): 363–372. doi:10.1145/358407.358414. S2CID 2634158. **^**Ye, Zhi-Sheng; Chen, Nan (2017). "Closed-Form Estimators for the Gamma Distribution Derived from Likelihood Equations".*The American Statistician*.**71**(2): 177–181. doi:10.1080/00031305.2016.1209129. S2CID 124682698. Archived from the original on 2023-05-26. Retrieved 2019-07-27.**^**Louzada, Francisco; Ramos, Pedro L.; Ramos, Eduardo (2019). "A Note on Bias of Closed-Form Estimators for the Gamma Distribution Derived from Likelihood Equations".*The American Statistician*.**73**(2): 195–199. doi:10.1080/00031305.2018.1513376. S2CID 126086375. Archived from the original on 2023-05-26. Retrieved 2019-07-27.**^**Fink, D. 1995 A Compendium of Conjugate Priors. In progress report: Extension and enhancement of methods for setting data quality objectives. (DOE contract 95‑831).**^**Jessica., Scheiner, Samuel M., 1956- Gurevitch (2001). "13. Failure-time analysis".*Design and analysis of ecological experiments*. Oxford University Press. ISBN 0-19-513187-8. OCLC 43694448. Archived from the original on 2024-10-09. Retrieved 2022-05-26.`{{cite book}}`

: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)**^**Golubev, A. (March 2016). "Applications and implications of the exponentially modified gamma distribution as a model for time variabilities related to cell proliferation and gene expression".*Journal of Theoretical Biology*.**393**: 203–217. Bibcode:2016JThBi.393..203G. doi:10.1016/j.jtbi.2015.12.027. ISSN 0022-5193. PMID 26780652. Archived from the original on 2024-10-09. Retrieved 2022-05-26.**^**Poon, Art; Davis, Bradley H; Chao, Lin (2005-07-01). "The Coupon Collector and the Suppressor Mutation".*Genetics*.**170**(3): 1323–1332. doi:10.1534/genetics.104.037259. ISSN 1943-2631. PMC 1451182. PMID 15879511. Archived from the original on 2024-10-09. Retrieved 2022-05-26.**^**Vineyard, Michael; Amoako-Gyampah, Kwasi; Meredith, Jack R (July 1999). "Failure rate distributions for flexible manufacturing systems: An empirical study".*European Journal of Operational Research*.**116**(1): 139–155. doi:10.1016/s0377-2217(98)00096-4. ISSN 0377-2217. Archived from the original on 2024-10-09. Retrieved 2022-05-26.**^**Rief, Matthias; Rock, Ronald S.; Mehta, Amit D.; Mooseker, Mark S.; Cheney, Richard E.; Spudich, James A. (2000-08-15). "Myosin-V stepping kinetics: A molecular model for processivity".*Proceedings of the National Academy of Sciences*.**97**(17): 9482–9486. Bibcode:2000PNAS...97.9482R. doi:10.1073/pnas.97.17.9482. ISSN 0027-8424. PMC 16890. PMID 10944217.**^**p. 43, Philip J. Boland, Statistical and Probabilistic Methods in Actuarial Science, Chapman & Hall CRC 2007**^**Wilks, Daniel S. (1990). "Maximum Likelihood Estimation for the Gamma Distribution Using Data Containing Zeros".*Journal of Climate*.**3**(12): 1495–1501. Bibcode:1990JCli....3.1495W. doi:10.1175/1520-0442(1990)003<1495:MLEFTG>2.0.CO;2. ISSN 0894-8755. JSTOR 26196366.**^**Belikov, Aleksey V. (22 September 2017). "The number of key carcinogenic events can be predicted from cancer incidence".*Scientific Reports*.**7**(1): 12170. Bibcode:2017NatSR...712170B. doi:10.1038/s41598-017-12448-7. PMC 5610194. PMID 28939880.**^**Belikov, Aleksey V.; Vyatkin, Alexey; Leonov, Sergey V. (2021-08-06). "The Erlang distribution approximates the age distribution of incidence of childhood and young adulthood cancers".*PeerJ*.**9**: e11976. doi:10.7717/peerj.11976. ISSN 2167-8359. PMC 8351573. PMID 34434669.**^**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)**^**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.**^**DJ Reiss, MT Facciotti and NS Baliga (2008) "Model-based deconvolution of genome-wide DNA binding",*Bioinformatics*, 24, 396–403**^**MA Mendoza-Parra, M Nowicka, W Van Gool, H Gronemeyer (2013) "Characterising ChIP-seq binding patterns by model-based peak shape deconvolution" Archived 2024-10-09 at the Wayback Machine,*BMC Genomics*, 14:834**^**Yang, Ziheng (September 1996). "Among-site rate variation and its impact on phylogenetic analyses".*Trends in Ecology & Evolution*.**11**(9): 367–372. Bibcode:1996TEcoE..11..367Y. CiteSeerX 10.1.1.19.99. doi:10.1016/0169-5347(96)10041-0. PMID 21237881. Archived from the original on 2024-04-12. Retrieved 2023-09-06.**^**Yang, Ziheng (September 1994). "Maximum likelihood phylogenetic estimation from DNA sequences with variable rates over sites: Approximate methods".*Journal of Molecular Evolution*.**39**(3): 306–314. Bibcode:1994JMolE..39..306Y. CiteSeerX 10.1.1.19.6626. doi:10.1007/BF00160154. ISSN 0022-2844. PMID 7932792. S2CID 17911050. Archived from the original on 2024-10-09. Retrieved 2023-09-06.**^**Felsenstein, Joseph (2001-10-01). "Taking Variation of Evolutionary Rates Between Sites into Account in Inferring Phylogenies".*Journal of Molecular Evolution*.**53**(4–5): 447–455. Bibcode:2001JMolE..53..447F. doi:10.1007/s002390010234. ISSN 0022-2844. PMID 11675604. S2CID 9791493. Archived from the original on 2024-10-09. Retrieved 2023-09-06.- ^
^{a}^{b}^{c}Devroye, Luc (1986).*Non-Uniform Random Variate Generation*. New York: Springer-Verlag. ISBN 978-0-387-96305-1. Archived from the original on 2012-07-17. Retrieved 2012-02-26. See Chapter 9, Section 3. - ^
^{a}^{b}Ahrens, J. H.; Dieter, U (January 1982). "Generating gamma variates by a modified rejection technique".*Communications of the ACM*.**25**(1): 47–54. doi:10.1145/358315.358390. S2CID 15128188.. See Algorithm GD, p. 53. **^**Ahrens, J. H.; Dieter, U. (1974). "Computer methods for sampling from gamma, beta, Poisson and binomial distributions".*Computing*.**12**(3): 223–246. CiteSeerX 10.1.1.93.3828. doi:10.1007/BF02293108. S2CID 37484126.**^**Cheng, R. C. H.; Feast, G. M. (1979). "Some Simple Gamma Variate Generators".*Journal of the Royal Statistical Society. Series C (Applied Statistics)*.**28**(3): 290–295. doi:10.2307/2347200. JSTOR 2347200.**^**Marsaglia, G. The squeeze method for generating gamma variates. Comput, Math. Appl. 3 (1977), 321–325.

The Wikibook *Statistics* has a page on the topic of: **Gamma distribution**

- "Gamma-distribution",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Weisstein, Eric W. "Gamma distribution".
*MathWorld*. - ModelAssist (2017) Uses of the gamma distribution in risk modeling, including applied examples in Excel Archived 2017-05-09 at the Wayback Machine.
- Engineering Statistics Handbook