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Encyclopedia > Gamma distribution
Gamma
Probability density function
Probability density plots of gamma distributions
Cumulative distribution function
Cumulative distribution plots of gamma distributions
Parameters k > 0, shape (real)
theta > 0, scale (real)
Support x in [0; infty)!
Probability density function (pdf) x^{k-1} frac{exp{left(-x/thetaright)}}{Gamma(k),theta^k},!
Cumulative distribution function (cdf) frac{gamma(k, x/theta)}{Gamma(k)},!
Mean k theta,!
Median no simple closed form
Mode (k-1) thetatext{ for }k geq 1,!
Variance k theta^2,!
Skewness frac{2}{sqrt{k}},!
Excess kurtosis frac{6}{k},!
Entropy k + lntheta + lnGamma(k) !
+ (1-k)psi(k) !
Moment-generating function (mgf) (1 - theta,t)^{-k}text{ for }t < 1/theta,!
Characteristic function (1 - theta,i,t)^{-k},!

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. It has a scale parameter θ and a shape parameter k. If k is an integer then the distribution represents the sum of k exponentially distributed random variables, each of which has parameter frac{1}{theta}. Download high resolution version (1300x975, 158 KB) See the image on the commons for gnuplot source. ... Image File history File links Download high resolution version (1300x975, 174 KB) Please see the file description page for further information. ... In probability theory and statistics, a shape parameter is a special kind of numerical parameter of a parametric family of probability distributions. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... This article is about the statistical concept. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... This article is about mathematics. ... Example of experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... This article is about the field of statistics. ... A probability distribution describes the values and probabilities that a random event can take place. ... In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... In probability theory and statistics, a shape parameter is a special kind of numerical parameter of a parametric family of probability distributions. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ...

Contents

Characterization

That a random variable X is gamma-distributed with scale θ and shape k is denoted

X sim Gamma(k, theta) ,,mathrm{ or },, X sim textrm{Gamma}(k, theta).

Probability density function

The probability density function of the gamma distribution can be expressed in terms of the gamma function: In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... The Gamma function along part of the real axis In mathematics, the Gamma function (represented by the capitalized Greek letter Γ) is an extension of the factorial function to real and complex numbers. ...

f(x;k,theta) = x^{k-1} frac{e^{-x/theta}}{theta^k , Gamma(k)} mathrm{ for } x > 0,, mathrm{ and },, k, theta > 0.

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


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:

g(x;alpha,beta) = x^{alpha-1} frac{beta^{alpha} , e^{-beta,x} }{Gamma(alpha)} mathrm{for} x > 0 ,!.
If α is a positive integer, then Γ(α) = (α − 1)!

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


Cumulative distribution function

The cumulative distribution function is the regularized gamma function, which can be expressed in terms of the incomplete gamma function, In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In mathematics, the gamma function is defined by a definite integral. ...

F(x;k,theta) = int_0^x f(u;k,theta),du =frac{gamma(k, x/theta)}{Gamma(k)} ,!

Properties

Summation

If Xi has a Γ(αi, β) distribution for i = 1, 2, ..., N, then

sum_{i=1}^N X_i sim Gamma left( sum_{i=1}^N alpha_i, beta right) ,!

provided all Xi are independent.


The gamma distribution exhibits infinite divisibility. The concept of infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). ...


Scaling

For any t > 0 it holds that tX is distributed Γ(k, tθ), demonstrating that θ is a scale parameter. In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ...


Exponential family

The Gamma distribution is a two-parameter exponential family with natural parameters k − 1 and − 1 / θ, and natural statistics X and ln(X). In probability and statistics, an exponential family is any class of probability distributions having a certain form. ... In probability and statistics, an exponential family is any class of probability distributions having a certain form. ... In probability and statistics, an exponential family is any class of probability distributions having a certain form. ...


Information entropy

The information entropy is given by Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ...

frac{-1}{theta^k Gamma(k)} int_0^{infty} frac{x^{k-1}}{e^{x/theta}} left[ (k-1)ln x - x/theta - k lntheta - lnGamma(k) right] ,dx !
= -left[ (k-1) (lntheta + psi(k)) - k - k lntheta - lnGamma(k) right] !
= k + lntheta + lnGamma(k) + (1-k)psi(k) !

where ψ(k) is the digamma function. In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions. ...


Kullback–Leibler divergence

The directed Kullback–Leibler divergence between Γ(α0, β0) ('true' distribution) and Γ(α, β) ('approximating' distribution) is given by In probability theory and information theory, the Kullback–Leibler divergence (or information divergence, or information gain, or relative entropy) is a natural distance measure from a true probability distribution P to an arbitrary probability distribution Q. Typically P represents data, observations, or a precise calculated probability distribution. ...

D_{mathrm{KL}}(alpha,beta || alpha_0, beta_0) = logleft(frac{Gamma({alpha_0})beta_0^{alpha_0}}{Gamma(alpha)beta^{alpha_0}}right)+(alpha-{alpha_0})psi(alpha)+alphafrac{beta-beta_0}{beta_0}

Laplace transform

The Laplace transformation of the gamma distribution is

F(s)=frac{beta^alpha}{(s+beta)^alpha}.

Parameter estimation

Maximum likelihood estimation

The likelihood function for N iid observations (x_1,ldots,x_N) is In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ...

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

from which we calculate the log-likelihood function

ell(theta) = (k-1) sum_{i=1}^N ln{(x_i)} - sum x_i/theta - Nkln{(theta)} - Nln{Gamma(k)}.

Finding the maximum with respect to θ by taking the derivative and setting it equal to zero yields the maximum likelihood estimate of the θ parameter: Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set. ...

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}^Nln{(x_i)}-Nk-Nkln{left(frac{sum x_i}{kN}right)}-Nln{(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_iright)}-frac{1}{N}sum_{i=1}^Nln{(x_i)} ,!

where psi(k) = frac{Gamma'(k)}{Gamma(k)} ! is the digamma function. In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions. ...


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 In numerical analysis, Newtons method (also known as the Newton–Raphson method or the Newton–Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ... In statistics, the method of moments is a method of estimation of population parameters such as mean, variance, median, etc. ...

ln(k)-psi(k) approx frac{1}{k}left(frac{1}{2} + frac{1}{12k+2}right). ,!

If we let

s = ln{left(frac{1}{N}sum_{i=1}^N x_iright)} - frac{1}{N}sum_{i=1}^Nln{(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 - psileft(kright) - s }{ 1/k - psi'left(kright) }

where psi'left(cdotright) 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:

psileft(kright) = begin{cases} ln(k) - ( 1 + ( 1 - ( 1/10 - 1 / ( 21 k^2 ) ) / k^2 ) / ( 6 k ) ) / ( 2 k ), quad k geq 8 psileft( k + 1 right) - 1/k, quad k < 8 end{cases}

and

psi'left(kright) = begin{cases} ( 1 + ( 1 + ( 1 - ( 1/5 - 1 / ( 7 k^2 ) ) / k^2 ) / ( 3 k ) ) / ( 2 k ) ) / 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 θ, the posterior PDF for theta (using the standard scale-invariant prior for θ) is

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

Denoting

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

int_0^{infty} theta^{-N k-1+m} e^{-y / theta}, dtheta = int_0^{infty} x^{N k -1 -m} e^{-x y} , dx = y^{-(N k -m)} Gamma(N k -m). !

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

E(x^m) = frac {Gamma (N k -m)} {Gamma(N k)} y^m, !

which shows that the mean +/- standard deviation estimate of the posterior distribution for theta is

frac {y} {N k -1} +/- frac {y^2} {(N k-1)^2 (N k-2)}.

←


Generating gamma-distributed random variables

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


Using the fact that a Γ(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 Γ(1, 1). Now, using the "α-addition" property of gamma distribution, we expand this result: In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distributions support are equally probable. ...

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

where Uk are all uniformly distributed on (0, 1] and independent.


All that is left now is to generate a variable distributed as Γ(δ, 1) for 0 < δ < 1 and apply the "α-addition" property once more. This is the most difficult part.


We provide an algorithm without proof. It is an instance of the acceptance-rejection method: In mathematics, rejection sampling is a technique used to generate observations from a distribution. ...

  1. Let m be 1.
  2. Generate V3m − 2, V3m − 1 and V3m — independent uniformly distributed on (0, 1] variables.
  3. If V_{3m - 2} le v_0, where v_0 = frac e {e + delta}, then go to step 4, else go to step 5.
  4. Let &# 0;m = V_{3m - 1}^{1 / delta}, eta_m = V_{3m} &# 0;_m^ {delta - 1}. Go to step 6.
  5. Let &# 0;m = 1 - ln {V_{3m - 1}}, eta_m = V_{3m} e^{-&# 0;m}.
  6. If eta_m > &# 0;m^{delta - 1} e^{-&# 0;m}, then increment m and go to step 2.
  7. Assume ξ = ξm to be the realization of Γ(δ,1)

Now, to summarize,

theta left( &# 0;- sum _{i=1} ^{[k]} {ln U_i} right) sim Gamma (k, theta),

where [k] is the integral part of k, and ξ has been generated using the algorithm above with δ = {k} (the fractional part of k), Uk and Vl are distributed as explained above and are all independent.


The GNU Scientific Library has robust routines for sampling many distributions including the Gamma distribution. Code using the library and the computed results In computing, GNU Scientific Library (or GSL) is a software library written in the C programming language for numerical calculations in applied mathematics and science. ...


Related distributions

Specializations

  • If X sim {Gamma}(k=1, theta=1/lambda),, then X has an exponential distribution with rate parameter λ.
  • If X sim {Gamma}(k=v/2, theta=2),, then X is identical to χ2(ν), the chi-square distribution with ν degrees of freedom.
  • 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 X^2 sim {Gamma}(3/2, 2a^2),, then X has a Maxwell-Boltzmann distribution with parameter a.
  • X sim mathrm{SkewLogistic}(theta),, then mathrm{log}(1 + e^{-X}) sim Gamma (1,theta),

In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... This article is about the mathematics of the chi-square distribution. ... The Erlang distribution is a continuous probability distribution with wide applicability primarily due to its relation to the exponential and Gamma distributions. ... It has been suggested that this article be split into multiple articles. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...

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.

The inverse gamma distribution has the probability density function over the support with shape parameter and scale parameter . ... In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]: where α and β are parameters that must be greater than zero and B is the beta function. ... Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors α. Clockwise from top left: α=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4). ...

References

Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

See also

  • Gamma distribution on wikibooks
 v • d • e 
Probability distributions
Discrete univariate with finite support
Discrete univariate with infinite support
Continuous univariate supported on a bounded interval
Continuous univariate supported on a semi-infinite interval
Beta prime · Burr · chi-square · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scale inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda
Continuous univariate supported on the whole real line
Multivariate (joint)
Directional, degenerate, and singular
Families
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In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. ... In probability theory, a Lévy skew alpha-stable distribution or just stable distribution, developed by Paul Lévy, is a probability distribution where sums of independent identically distributed random variables have the same distribution as the original. ... In probability theory and statistics, the logistic distribution is a continuous probability distribution. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In probability theory and statistics, the normal-gamma distribution is a four-parameter family of continuous probability distributions. ... The normal-inverse Gaussian distribution is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. ... In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ... In probability theory, the Type-1 Gumbel distribution function is for . Reference Taken from the gsl-ref_19. ... The variance-gamma distribution is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the gamma distribution. ... In spectroscopy, the Voigt profile is a spectral line profile named after Woldemar Voigt and found in all branches of spectroscopy in which a spectral line is broadened by two types of mechanisms, one of which alone would produce a Doppler profile, and the other of which would produce a... A probability distribution describes the values and probabilities that a random event can take place. ... In population genetics, Ewenss sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are a1 alleles represented once... In probability theory, the multinomial distribution is a generalization of the binomial distribution. ... The multivariate Polya distribution, also called the Dirichlet compound multinomial distribution, is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution and a set of discrete samples x is drawn from the multinomial distribution with probability vector p. ... Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors α. Clockwise from top left: α=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4). ... In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and twice the number of parameters. ... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ... In statistics, a multivariate Student distribution is a multivariate generalization of the Students t-distribution. ... A probability distribution describes the values and probabilities that a random event can take place. ... In statistics, the Inverse Wishart distribution, also the inverse Wishart distribution and inverted Wishart distribution is a probability density function defined on matrices. ... The matrix normal distribution is a probability distribution that is a generalization of the normal distribution. ... In statistics, the Wishart distribution, named in honor of John Wishart, is any of a family of probability distributions for nonnegative-definite matrix-valued random variables (random matrices). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. ... Circular or directional statistics is the subdiscipline of statistics that deals with circular or directional data. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... In probability, a singular distribution is a probability distribution concentrated on a measure zero set where the probability of each point in that set is zero. ... Circular or directional statistics is the subdiscipline of statistics that deals with circular or directional data. ... The 5-parameter Fisher-Bingham distribution or Kent distribution is a probability distribution on the three-dimensional sphere. ... In probability theory and statistics, the von Mises distribution is a continuous probability distribution. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. ... In probability, a singular distribution is a probability distribution concentrated on a measure zero set where the probability of each point in that set is zero. ... The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. ... In probability and statistics, an exponential family is any class of probability distributions having a certain form. ... In probability theory, especially as that field is used in statistics, a location-scale family is a set of probability distributions on the real line parametrized by a location parameter μ and a scale parameter σ â‰¥ 0; if X is any random variable whose probability distribution belongs to such a family, then... In statistics and information theory, a maximum entropy probability distribution is a probability distribution whose entropy is larger than (or equal to) that of all other members of a specified class of distributions. ... This article or section is incomplete and may require expansion and/or cleanup. ...

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BioMed Central | Full text | Statistical distributions of optimal global alignment scores of random protein sequences (4036 words)
The normal distribution also agrees well with the score distribution frequencies when the shape parameter of the gamma distribution is sufficiently large, for this is the scenario when the normal distribution can be viewed as an approximation of the gamma distribution.
Both the gamma and normal distributions fit the score frequencies of the former well (Figure 3), whereas for the latter, the normal distribution disagrees with the majority of the score curve.
The score distribution was fitted with (A) the three-parameter gamma distribution; (B) the normal distribution; and (C) the Gumbel distribution.
Random Number Generator - Gamma Distribution (173 words)
The Gamma distribution is most often used to describe the distribution of the amount of time until the nth occurrence of an event in a Poisson process.
For example, when a Gamma distribution has an alpha of 1, Gamma(1, b), it becomes an Exponential distribution with scale parameter of b, Expo(b).
And a Chi-Square distribution with k df is the same as the Gamma(k/2, 2) distribution.
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