Gamma distribution

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. There are two different parameterizations in common use:

With a shape parameter k and a scale parameter θ.
With a shape parameter α = k and an inverse scale parameter β = 1⁄θ, called a rate parameter.

The parameterization with k and θ appears to be more common in econometrics and certain other applied fields, where e.g. 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.[1]

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 (aka rate) parameters, such as the λ of an exponential distribution or a Poisson distribution — 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 an 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 (which is equivalent to a rate parameter of 1/θ). Equivalently, if α is an integer, then the distribution again represents an Erlang distribution, i.e. the sum of α independent exponentially-distributed random variables, each of which has a mean of 1/β (which is equivalent to a rate parameter of β).

The gamma distribution is the maximum entropy probability distribution for a random variable X for which \( \scriptstyle E(X) \;=\; k\theta \;=\; \alpha/\beta \) is fixed and greater than zero, and \( \scriptstyle E[\ln(X)] \;=\; \psi(k) \,+\, \ln(\theta) \;=\; \psi(\alpha) \,-\, \ln(\beta) is fixed (\scriptstyle \psi \) is the digamma function).[2]

Characterization using shape k and scale θ

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

\( X \sim \Gamma(k, \theta) \equiv \textrm{Gamma}(k,\theta ) \)

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

The probability density function of the gamma distribution can be expressed in terms of the gamma function parameterized in terms of a shape parameter k and scale parameter θ. Both k and θ will be positive values.

The equation defining the probability density function of a gamma-distributed random variable x is

\( \begin{align} f(x;k,\theta) &= \frac{1}{\theta^k}\frac{1}{\Gamma(k)}x^{k-1}e^{-\frac{x}{\theta}} \\ & \text{ for } x \geq 0 \text{ and } k, \theta > 0 \end{align} \)

(This parameterization is used in the infobox and the plots.)
Cumulative distribution function

The cumulative distribution function is the regularized gamma function:

\( F(x;k,\theta) = \int_0^x f(u;k,\theta)\,du = \frac{\gamma\left(k, \frac{x}{\theta}\right)}{\Gamma(k)} \, \)

where \scriptstyle \gamma(k,\, x/\theta) 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):[3]

\( F(x;k,\theta) = 1-\sum_{i=0}^{k-1} \frac{1}{i!} \left(\frac{x}{\theta}\right)^i e^{-\frac{x}{\theta}} = \sum_{i=k}^{\infty} \frac{1}{i!} \left(\frac{x}{\theta}\right)^i e^{-\frac{x}{\theta}} \)

Characterization using shape α and rate β

Alternatively, 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:

\( \begin{align} g(x;\alpha,\beta) &= \beta^{\alpha}\frac{1}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x} \\ & \text{ for } x \geq 0 \text{ and } \alpha, \beta > 0 \end{align} \)

If α is a positive integer, then

\( \Gamma(\alpha) = (\alpha - 1)!\, \)

A random variable X that is gamma-distributed with shape α and scale β is denoted

\( X \sim \Gamma(\alpha, \beta) \equiv \textrm{Gamma}(\alpha,\beta) \)

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

The cumulative distribution function is the regularized gamma function:

\( F(x;\alpha,\beta) = \int_0^x f(u;\alpha,\beta)\,du = \frac{\gamma(\alpha, \beta x)}{\Gamma(\alpha)} \, \)

where \( \scriptstyle \gamma(\alpha,\, \beta x) \) is the lower incomplete gamma function.

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

\( F(x;\alpha,\beta) = 1-\sum_{i=0}^{\alpha-1} \frac{1}{i!} (\beta x)^i e^{-\beta x} = \sum_{i=\alpha}^{\infty} \frac{1}{i!} (\beta x)^i e^{-\beta x} \)

Properties
Median calculation

Unlike the mode and the mean which have readily calculable formulas based on the parameters, the median does not have an easy closed form equation. The median for this distribution is defined as the constant x0 such that

\( {1 \over \Gamma(k) \theta^k} \int_0^{x_0} x^{k-1} e^{-\frac{x}{\theta}} dx = {1 \over 2} \)

The ease of this calculation is dependent on the k parameter. This is best achieved by a computer since the calculations can quickly grow out of control.
Summation

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

\( \sum_{i=1}^N X_i \sim \mathrm{Gamma} \left( \sum_{i=1}^N k_i, \theta \right) \,\! \)

provided all Xi' are independent.

The gamma distribution exhibits infinite divisibility.

Scaling

If

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

then for any c > 0,

\( cX \sim \mathrm{Gamma}( k, c\theta) \, \)

Hence the use of the term "scale parameter" to describe θ.

Equivalently, if

\( X\,\sim\,\mathrm{Gamma}(\alpha, \beta) \, \)

then for any c > 0,

\( cX \sim \mathrm{Gamma}( \alpha, \beta/c) \, \)

Hence the use of the term "inverse scale parameter" to describe β.
Exponential family

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

One can show that

\( \mathbb{E}[\ln(x)] = \psi(\alpha) - \ln(\beta) \)

or equivalently,

\(\mathbb{E}[\ln(x)] = \psi(k) + \ln(\theta) \)

where ψ(α) or ψ(k) is the digamma function.

This can be derived using the exponential family formula for the moment generating function of the sufficient statistic, because one of the sufficient statistics of the gamma distribution is \ln x .
Information entropy

The information entropy can be derived as

\( \begin{align} H(x) = \operatorname{E}(-\ln p(x)) &= \operatorname{E}(-\alpha\ln\beta + \ln\Gamma(\alpha) - (\alpha-1)\ln x + \beta x) \\ &= -\alpha\ln\beta + \ln\Gamma(\alpha) - (\alpha-1)\operatorname{E}(\ln x) + \beta \operatorname{E}(x) \\ &= -\alpha\ln\beta + \ln\Gamma(\alpha) - (\alpha-1)(\psi(\alpha) - \ln(\beta)) + \beta \frac{\alpha}{\beta} \\ &= -\alpha\ln\beta + \ln\Gamma(\alpha) - (\alpha-1)\psi(\alpha) +\alpha\ln\beta - \ln\beta + \alpha \\ &= \ln\Gamma(\alpha) - (\alpha-1)\psi(\alpha) - \ln\beta + \alpha \\ &= \alpha - \ln\beta + \ln\Gamma(\alpha) + (1-\alpha)\psi(\alpha) \end{align} \)

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

\( k + \ln\theta + \ln\Gamma(k) + (1-k)\psi(k) \)

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

The Kullback–Leibler divergence (KL-divergence), as with the information entropy and various other theoretical properties, are more commonly seen using the α,β parameterization because of their uses in Bayesian and other theoretical statistics frameworks.

The KL-divergence of \( \rm{Gamma}(\alpha_p, \beta_p) \) ("true" distribution) from \( \rm{Gamma}(\alpha_q, \beta_q) \) ("approximating" distribution) is given by[5]

\( \begin{align} D_{\mathrm{KL}}(\alpha_p,&\,\beta_p; \alpha_q, \beta_q) = \\ & (\alpha_p-\alpha_q)\psi(\alpha_p) - \log\Gamma(\alpha_p) + \log\Gamma(\alpha_q) \\ & + \alpha_q(\log \beta_p - \log \beta_q) + \alpha_p\frac{\beta_q-\beta_p}{\beta_p} \end{align} \)

Written using the k,θ parameterization, the KL-divergence of \( \rm{Gamma}(k_p, \theta_p) \) from \( \rm{Gamma}(k_q, \theta_q) \) is given by

\( D_{\mathrm{KL}}(k_p,\theta_p; k_q, \theta_q) = (k_p-k_q)\psi(k_p) - \log\Gamma(k_p) + \log\Gamma(k_q) + k_q(\log \theta_q - \log \theta_p) + k_p\frac{\theta_p - \theta_q}{\theta_q} \)

Laplace transform

The Laplace transform of the gamma PDF is

\( F(s) = \left(1 + \theta s\right)^{-k} = \frac{\beta^\alpha}{(s + \beta)^\alpha} \)

Parameter estimation
Maximum likelihood estimation

The likelihood function for N iid observations (x1, ..., xN) is

\( L(k, \theta) = \prod_{i=1}^N f(x_i;k,\theta)\,\! \)

from which we calculate the log-likelihood function

\(\ell(k, \theta) = (k - 1) \sum_{i=1}^N \ln{(x_i)} - \sum_{i=1}^N \frac{x_i}{\theta} - Nk\ln{(\theta)} - N\ln{\Gamma(k)} \)

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:

\( \hat{\theta} = \frac{1}{kN}\sum_{i=1}^N x_i \,\! \)

Substituting this into the log-likelihood function gives

\( \ell = (k-1)\sum_{i=1}^N\ln{(x_i)} - Nk - Nk\ln{\left(\frac{\sum x_i}{kN}\right)} - N\ln[\Gamma(k)] \,\! \)

Finding the maximum with respect to k by taking the derivative and setting it equal to zero yields

\( \ln{(k)} - \psi(k) = \ln{\left(\frac{1}{N}\sum_{i=1}^N x_i\right)} - \frac{1}{N}\sum_{i=1}^N\ln{(x_i)} \,\! \)

where

\( \psi(k) = \frac{\Gamma'(k)}{\Gamma(k)} \! \)

is the digamma function.

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

\( \ln(k) - \psi(k) \approx \frac{1}{2k}\left(1 + \frac{1}{6k + 1}\right) \,\! \)

If we let

\( s = \ln{\left(\frac{1}{N}\sum_{i=1}^N x_i\right)} - \frac{1}{N}\sum_{i=1}^N\ln{(x_i)}\,\! \)

then k is approximately

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

which is within 1.5% of the correct value.[citation needed] An explicit form for the Newton-Raphson update of this initial guess is given by Choi and Wette (1969) as the following expression:

\( k \leftarrow k - \frac{ \ln k - \psi\left(k\right) - s }{ \frac{1}{k} - \psi\;'\left(k\right) } \)

where \psi' denotes the trigamma function (the derivative of the digamma function).

The digamma and trigamma functions can be difficult to calculate with high precision. However, approximations known to be good to several significant figures can be computed using the following approximation formulae:

\( \psi\left(k\right) = \begin{cases} \ln(k) - \left( 1 + \left[ 1 - \left( 1/10 - 1 / 21k^2 \right) / k^2 \right] / 6k \right) / 2k , \quad k \geq 8 \\ \psi\left( k + 1 \right) - 1/k, \quad k < 8 \end{cases} \)

and

\( \psi\;'\left(k\right) = \begin{cases} ( 1 + [ 1 + ( 1 - [ 1/5 - 1 / 7k^2 ] / k^2 ) / 3k ] / 2k ) / k, \quad k \geq 8, \\ \psi\;'\left( k + 1 \right) + 1/k^2, \quad k < 8 \end{cases} \)

For details, see Choi and Wette (1969).
Bayesian minimum mean-squared error

With known k and unknown \theta , the posterior PDF for theta (using the standard scale-invariant prior for \theta) is

\( P(\theta | k, x_1, \dots, x_N) \propto 1/\theta \prod_{i=1}^N f(x_i; k, \theta)\,\! \)

Denoting

\( y \equiv \sum_{i=1}^Nx_i , \qquad P(\theta | k, x_1, \dots, x_N) = C(x_i) \theta^{-N k-1} e^{-\frac{y}{\theta}}\! \)

Integration over θ can be carried out using a change of variables, revealing that 1⁄θ is gamma-distributed with parameters \( \scriptstyle \alpha \;=\; Nk,\; \beta \;=\; y. \)

\( \int_0^{\infty} \theta^{-Nk - 1 + m} e^{-\frac{y}{\theta}}\, d\theta = \int_0^{\infty} x^{Nk - 1 - m} e^{-xy} \, dx = y^{-(Nk - m)} \Gamma(Nk - m) \! \)

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

\( E(x^m) = \frac {\Gamma (Nk - m)} {\Gamma(Nk)} y^m \! \)

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

\( \frac {y} {Nk - 1} \pm \frac {y^2} {(Nk - 1)^2 (Nk - 2)} \)

Generating gamma-distributed random variables

Given the scaling property above, it is enough to generate gamma variables with \( \scriptstyle \theta \;=\; 1 \) as we can later convert to any value of \scriptstyle \beta with simple division.

Using the fact that a \( \scriptstyle \Gamma(1,\, 1) \) distribution is the same as an \( \scriptstyle Exp(1)\) distribution, and noting the method of generating exponential variables, we conclude that if \scriptstyle U is uniformly distributed on \(\scriptstyle (0,\, 1] \), then −\ln(U) is distributed \( \scriptstyle \Gamma(1,\, 1) \) Now, using the "α-addition" property of gamma distribution, we expand this result:

\( \sum_{k=1}^n {-\ln U_k} \sim \Gamma(n, 1) \)

where \( \scriptstyle U_k \) are all uniformly distributed on \( \scriptstyle (0,\, 1] \) and independent. All that is left now is to generate a variable distributed as \( \scriptstyle \Gamma(\delta,\, 1) \) for \( \scriptstyle 0 \;<\; \delta \;<\; 1 \) 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,[6] noting that none are uniformly fast for all shape parameters. For small values of the shape parameter, the algorithms are often not valid.[7] For arbitrary values of the shape parameter, one can apply the Ahrens and Dieter[8] modified acceptance-rejection method Algorithm GD (shape k ≥ 1), or transformation method[9] when 0 < k < 1. Also see Cheng and Feast Algorithm GKM 3[10] or Marsaglia's squeeze method.[11]

The following is a version of the Ahrens-Dieter acceptance-rejection method:[8]

Let \( \scriptstyle m \) be 1.
Generate \( \scriptstyle V_{3m - 2}, \scriptstyle V_{3m - 1} \) and \( \scriptstyle V_{3m} \) as independent uniformly distributed on \( \scriptstyle (0,\, 1] \) variables.
If \( \scriptstyle V_{3m - 2} \;\le\; v_0 \), where \( \scriptstyle v_0 \;=\; \frac e {e \;+\; \delta} \) , then go to step 4, else go to step 5.
Let \( \scriptstyle \xi_m \;=\; V_{3m - 1}^{1 / \delta}, \ \eta_m \;=\; V_{3m} \xi _m^ {\delta - 1} \). Go to step 6.
Let \( \scriptstyle \( \xi_m \;=\; 1 \,-\, \ln {V_{3m - 1}}, \ \eta_m \;=\; V_{3m} e^{-\xi_m}. \)
If \( \scriptstyle \eta_m \;>\; \xi_m^{\delta - 1} e^{-\xi_m} \), then increment \scriptstyle m and go to step 2.
Assume \( \scriptstyle \xi \;=\; \xi_m \) to be the realization of \( \scriptstyle \Gamma (\delta,\, 1) \).

A summary of this is

\( \theta \left( \xi - \sum _{i=1} ^{\lfloor{k}\rfloor} {\ln U_i} \right) \sim \Gamma (k, \theta) \)

where

\( \scriptstyle \lfloor{k}\rfloor \) is the integral part of \( \scriptstyle k, \)
\(\scriptstyle \xi \) has been generated using the algorithm above with \( \scriptstyle \delta \;=\; \{k\} \) (the fractional part of \( \scriptstyle k \)),
\( \scriptstyle U_k \) and \( \scriptstyle V_l \) are distributed as explained above and are all independent.

Related distributions
Specializations

If \( \scriptstyle X \;\sim\; {\Gamma}(k \;=\; 1,\, \theta \;=\; 1/\lambda)\,,\) then X has an exponential distribution with rate parameter λ.
If \( \scriptstyle X \;\sim\; {\Gamma}(k \;=\; \nu/2,\, \theta \;=\; 2)\,,\) then X is identical to χ2(ν), the chi-squared distribution with ν degrees of freedom. Conversely, if \( \scriptstyle Q \;\sim\; {\chi}^2(\nu)\,\) and c is a positive constant, then \( \scriptstyle c \cdot Q \;\sim\; {\Gamma}(k \;=\; \nu/2,\, \theta \;=\; 2c)\, \).
If \( \scriptstyle k \) is an integer, the gamma distribution is an Erlang distribution and is the probability distribution of the waiting time until the \( \scriptstyle k-th \) "arrival" in a one-dimensional Poisson process with intensity 1/θ. If \( \scriptstyle X \;\sim\; {\Gamma}(k \;\in\; \mathbb{Z},\, \theta) \) and \( \scriptstyle Y \;\sim\; \mathrm{Pois}\left(\frac{x}{\theta}\right) \) , then \( \scriptstyle P(X \,>\, x) \;=\; P(Y \,<\, k). \)
If X has a Maxwell-Boltzmann distribution with parameter a, then \( \scriptstyle X^2 \;\sim\; {\Gamma}\left(\frac{3}{2},\, 2a^2\right)\, \).
\( \scriptstyle X \;\sim\; {\Gamma}(k,\, \theta)\, \), then \( \scriptstyle \sqrt{X}\ \), follows a generalized gamma distribution with parameters \( \scriptstyle p \;=\; 2, \scriptstyle d \;=\; 2k \), and \( \scriptstyle a \;=\; \sqrt{\theta} \).
\( \scriptstyle X \;\sim\; \mathrm{SkewLogistic}(\theta)\, \), then \( \scriptstyle \log\left(1 \,+\, e^{-X}\right) \;\sim\; \Gamma (1,\, \theta)\, \); i.e. an exponential distribution: see skew-logistic distribution.

Conjugate prior

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:[12]

\( p(k,\theta | p, q, r, s) = \frac{1}{Z} \frac{p^{k-1} e^{-\theta^{-1} q}}{\Gamma(k)^r \theta^{k s}} \)

Where Z is the normalizing constant, which has no closed form solution. The posterior distribution can be found by updating the parameters as follows.

\( \begin{align} p' &= p\prod_i x_i\\ q' &= q + \sum_i x_i\\ r' &= r + n\\ s' &= s + n \end{align} \)

Where \scriptstyle n is the number of observations, and \( \scriptstyle x_i is the \scriptstyle i^{th} \) observation.
Compound gamma

If the shape parameter of the gamma distribution is known, but the inverse-scale parameter is unknown, then a gamma distribution for the inverse-scale forms a conjugate prior. The compound distribution, which results from integrating out the inverse-scale has a closed form solution, known as the compound gamma distribution.[13]
Others

If X has a Γ(k, θ) distribution, then 1/X has an inverse-gamma distribution with parameters k and θ−1.
If X and Y are independently distributed Γ(α, θ) and Γ(β, θ) respectively, then X / (X + Y) has a beta distribution with parameters α and β.
If Xi are independently distributed Γ(αi,θ) respectively, then the vector (X1 / S, ..., Xn / S), where S = X1 + ... + Xn, follows a Dirichlet distribution with parameters α1, …, αn.
For large k the gamma distribution converges to Gaussian distribution with mean \scriptstyle \mu \;=\; k\theta and variance \( \scriptstyle \sigma^2 \;=\; k\theta^2. \)
The Gamma distribution is the conjugate prior for the precision of the normal distribution with known mean.
The Wishart distribution is a multivariate generalization 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 analogue of the Gamma distribution

Applications

The gamma distribution has been used to model the size of insurance claims[citation needed] and rainfalls.[14] This means that aggregate insurance claims and the amount of rainfall accumulated in a reservoir are modelled by a gamma process. The gamma distribution is also used to model errors in multi-level Poisson regression models, because the combination of the Poisson distribution and a gamma distribution is a negative binomial distribution.

In neuroscience, the gamma distribution is often used to describe the distribution of inter-spike intervals.[15] Although the gamma distribution often provides phenomenologically a good fit, there is no underlying biophysical motivation of it.

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.[16]

The gamma distribution is widely used as a conjugate prior in Bayesian statistics. 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.
See also

Gamma process
Lukacs's proportion-sum independence theorem
Chi-squared distribution
F-distribution
Hotelling's T-squared distribution
Student's t-distribution
Wilks' lambda distribution
Wishart distribution
Natural exponential family (with a known shape parameter α)
Exponential family

Notes

^ See Hogg and Craig (1978, Remark 3.3.1) for an explicit motivation
^ Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model". Journal of Econometrics (Elsevier): 219–230. Retrieved 2011-06-02.
^ Papoulis, Pillai, Probability, Random Variables, and Stochastic Processes, Fourth Edition
^ Papoulis, Pillai, Probability, Random Variables, and Stochastic Processes, Fourth Edition
^ W.D. Penny, KL-Divergences of Normal, Gamma, Dirichlet, and Wishart densities
^ Luc Devroye (1986). Non-Uniform Random Variate Generation. New York: Springer-Verlag. See Chapter 9, Section 3, pages 401–428.
^ Devroye (1986), p. 406.
^ a b Ahrens, J. H. and Dieter, U. (1982). Generating gamma variates by a modified rejection technique. Communications of the ACM, 25, 47–54. Algorithm GD, p. 53.
^ Ahrens, J. H. and Dieter, U. (1974). Computer methods for sampling from gamma, beta, Poisson and binomial distributions. Computing, 12, 223–246. PDF
^ Cheng, R.C.H., and Feast, G.M. Some simple gamma variate generators. Appl. Stat. 28 (1979), 290-295.
^ Marsaglia, G. The squeeze method for generating gamma variates. Comput, Math. Appl. 3 (1977), 321-325.
^ 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).
^ Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika 16: 27–31. doi:10.1007/BF02613934.
^ Aksoy, H. (2000) "Use of Gamma Distribution in Hydrological Analysis", Turk J. Engin Environ Sci, 24, 419 – 428.
^ 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)
^ 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.

References

R. V. Hogg and A. T. Craig. (1978) Introduction to Mathematical Statistics, 4th edition. New York: Macmillan. (See Section 3.3.)'
S. C. Choi and R. Wette. (1969) Maximum Likelihood Estimation of the Parameters of the Gamma Distribution and Their Bias, Technometrics, 11(4) 683–690

External links
The Wikibook Statistics has a page on the topic of
Gamma distribution

Weisstein, Eric W., "Gamma distribution" from MathWorld.
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