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The gamma distribution is a two-parameter family of curves. The gamma distribution models sums of exponentially distributed random variables and generalizes both the chi-square and exponential distributions.

Statistics and Machine Learning Toolbox™ offers several ways to work with the gamma distribution.

Create a probability distribution object

`GammaDistribution`

by fitting a probability distribution to sample data (`fitdist`

) or by specifying parameter values (`makedist`

). Then, use object functions to evaluate the distribution, generate random numbers, and so on.Work with the gamma distribution interactively by using the

**Distribution Fitter**app. You can export an object from the app and use the object functions.Use distribution-specific functions (

`gamcdf`

,`gampdf`

,`gaminv`

,`gamlike`

,`gamstat`

,`gamfit`

,`gamrnd`

,`randg`

) with specified distribution parameters. The distribution-specific functions can accept parameters of multiple gamma distributions.Use generic distribution functions (

`cdf`

,`icdf`

,`pdf`

,`random`

) with a specified distribution name (`'Gamma'`

) and parameters.

The gamma distribution uses the following parameters.

Parameter | Description | Support |
---|---|---|

`a`
| Shape | a` > 0` |

`b` | Scale | b` > 0` |

The standard gamma distribution has unit scale.

The sum of two gamma random variables with shape parameters
*a*_{1} and
*a*_{2} both with scale parameter
*b* is a gamma random variable with shape parameter *a* = *a*_{1}
+ *a*_{2} and scale parameter *b*.

The *likelihood function* is the probability density
function (pdf) viewed as a function of the parameters. The *maximum
likelihood estimates* (MLEs) are the parameter estimates that
maximize the likelihood function for fixed values of
`x`

.

The maximum likelihood estimators of *a* and
*b* for the gamma distribution are the solutions to the
simultaneous equations

$$\begin{array}{c}\mathrm{log}\widehat{a}-\psi (\widehat{a})=\mathrm{log}\left(\overline{x}/{\left({\displaystyle \prod _{i=1}^{n}{x}_{i}}\right)}^{1/n}\right)\\ \widehat{b}=\frac{\overline{x}}{\widehat{a}}\end{array}$$

where $$\overline{x}$$ is the sample mean for the sample *x*_{1},
*x*_{2}, …,
*x*_{n}, and *Ψ* is the digamma function `psi`

.

To fit the gamma distribution to data and find parameter estimates, use
`gamfit`

, `fitdist`

, or `mle`

. Unlike
`gamfit`

and `mle`

, which return
parameter estimates, `fitdist`

returns the fitted probability
distribution object `GammaDistribution`

. The object
properties `a`

and `b`

store the parameter
estimates.

For an example, see Fit Gamma Distribution to Data.

The pdf of the gamma distribution is

$$y=f(x|a,b)=\frac{1}{{b}^{a}\Gamma (a)}{x}^{a-1}{e}^{\frac{-x}{b}},$$

where Γ( · ) is the Gamma function.

For an example, see Compute Gamma Distribution pdf.

The cumulative distribution function (cdf) of the gamma distribution is

$$p=F(x|a,b)=\frac{1}{{b}^{a}\Gamma (a)}{\displaystyle \underset{0}{\overset{x}{\int}}{t}^{a-1}{e}^{\frac{-t}{b}}dt}.$$

The result *p* is the probability that a single observation from
the gamma distribution with parameters *a* and *b*
falls in the interval [0 *x*].

For an example, see Compute Gamma Distribution cdf.

The gamma cdf is related to the incomplete gamma function `gammainc`

by

$$f\left(x|a,b\right)=\text{gammainc}\left(\frac{x}{b},a\right).$$

The inverse cumulative distribution function (icdf) of the gamma distribution in terms of the gamma cdf is

$$x={F}^{-1}(p|a,b)=\{x:F(x|a,b)=p\},$$

where

$$p=F(x|a,b)=\frac{1}{{b}^{a}\Gamma (a)}{\displaystyle \underset{0}{\overset{x}{\int}}{t}^{a-1}{e}^{\frac{-t}{b}}dt}.$$

The result *x* is the value such that an observation from the
gamma distribution with parameters *a* and *b*
falls in the range [0 *x*] with
probability *p*.

The preceding integral equation has no known analytical solution. `gaminv`

uses an iterative approach
(Newton's method) to converge on the solution.

The mean of the gamma distribution is
*a**b*.

The variance of the gamma distribution is
*a**b ^{2}*.

Generate a sample of `100`

gamma random numbers with shape `3`

and scale `5`

.

x = gamrnd(3,5,100,1);

Fit a gamma distribution to data using `fitdist`

.

`pd = fitdist(x,'gamma')`

pd = GammaDistribution Gamma distribution a = 2.7783 [2.1374, 3.61137] b = 5.73438 [4.30198, 7.64372]

`fitdist`

returns a `GammaDistribution`

object. The intervals next to the parameter estimates are the 95% confidence intervals for the distribution parameters.

Estimate the parameters `a`

and `b`

using the distribution functions.

`[muhat,muci] = gamfit(x) % Distribution specific function`

`muhat = `*1×2*
2.7783 5.7344

`muci = `*2×2*
2.1374 4.3020
3.6114 7.6437

[muhat2,muci2] = mle(x,'distribution','gamma') % Generic function

`muhat2 = `*1×2*
2.7783 5.7344

`muci2 = `*2×2*
2.1374 4.3020
3.6114 7.6437

Compute the pdfs of the gamma distribution with several shape and scale parameters.

x = 0:0.1:50; y1 = gampdf(x,1,10); y2 = gampdf(x,3,5); y3 = gampdf(x,6,4);

Plot the pdfs.

figure; plot(x,y1) hold on plot(x,y2) plot(x,y3) hold off xlabel('Observation') ylabel('Probability Density') legend('a = 1, b = 10','a = 3, b = 5','a = 6, b = 4')

Compute the cdfs of the gamma distribution with several shape and scale parameters.

x = 0:0.1:50; y1 = gamcdf(x,1,10); y2 = gamcdf(x,3,5); y3 = gamcdf(x,6,4);

Plot the cdfs.

figure; plot(x,y1) hold on plot(x,y2) plot(x,y3) hold off xlabel('Observation') ylabel('Cumulative Probability') legend('a = 1, b = 10','a = 3, b = 5','a = 6, b = 4',"Location","northwest")

The gamma distribution has the shape parameter $\mathit{a}$ and the scale parameter $\mathit{b}$. For a large $\mathit{a}$, the gamma distribution closely approximates the normal distribution with mean $\mu =\mathit{ab}$ and variance ${\sigma}^{2}=\mathit{a}{\mathit{b}}^{2}$.

Compute the pdf of a gamma distribution with parameters `a = 100`

and `b = 5`

.

a = 100; b = 5; x = 250:750; y_gam = gampdf(x,a,b);

For comparison, compute the mean, standard deviation, and pdf of the normal distribution that gamma approximates.

mu = a*b

mu = 500

sigma = sqrt(a*b^2)

sigma = 50

y_norm = normpdf(x,mu,sigma);

Plot the pdfs of the gamma distribution and the normal distribution on the same figure.

plot(x,y_gam,'-',x,y_norm,'-.') title('Gamma and Normal pdfs') xlabel('Observation') ylabel('Probability Density') legend('Gamma Distribution','Normal Distribution')

The pdf of the normal distribution approximates the pdf of the gamma distribution.

Beta Distribution — The beta distribution is a two-parameter continuous distribution that has parameters

*a*(first shape parameter) and*b*(second shape parameter). If*X*_{1}and*X*_{2}have standard gamma distributions with shape parameters*a*_{1}and*a*_{2}respectively, then $$Y=\frac{{X}_{1}}{{X}_{1}\text{}+{X}_{2}}$$ has a beta distribution with shape parameters*a*_{1}and*a*_{2}.Chi-Square Distribution — The chi-square distribution is a one-parameter continuous distribution that has parameter

*ν*(degrees of freedom). The chi-square distribution is equal to the gamma distribution with*2a*=*ν*and*b*=*2*.Exponential Distribution — The exponential distribution is a one-parameter continuous distribution that has parameter

*μ*(mean). The exponential distribution is equal to the gamma distribution with*a*= 1 and*b*=*μ*. The sum of*k*exponentially distributed random variables with mean*μ*is the gamma distribution with parameters*a*=*k*and*μ*=*b*.Nakagami Distribution — The Nakagami distribution is a two-parameter continuous distribution with shape parameter

*µ*and scale parameter*ω*. If*x*has a Nakagami distribution, then*x*^{2}has a gamma distribution with*a*=*μ*and*a**b*=*ω*.Normal Distribution — The normal distribution is a two-parameter continuous distribution that has parameters

*μ*(mean) and*σ*(standard deviation). When*a*is large, the gamma distribution closely approximates a normal distribution with*μ*=*a**b*and*σ*=^{2}*a**b*. For an example, see Compare Gamma and Normal Distribution pdfs.^{2}

[1] Abramowitz, Milton, and
Irene A. Stegun, eds. *Handbook of Mathematical Functions: With Formulas,
Graphs, and Mathematical Tables*. 9. Dover print.; [Nachdr. der Ausg.
von 1972]. Dover Books on Mathematics. New York, NY: Dover Publ, 2013.

[2] Evans, Merran, Nicholas
Hastings, and Brian Peacock. *Statistical Distributions*. 2nd
ed. New York: J. Wiley, 1993.

[3] Hahn, Gerald J., and Samuel S. Shapiro. *Statistical Models in
Engineering*. Wiley Classics Library. New York: Wiley,
1994.

[4] Lawless, Jerald F.
*Statistical Models and Methods for Lifetime Data*. 2nd ed.
Wiley Series in Probability and Statistics. Hoboken, N.J: Wiley-Interscience,
2003.

[5] Meeker, William Q., and
Luis A. Escobar. *Statistical Methods for Reliability Data*.
Wiley Series in Probability and Statistics. Applied Probability and Statistics
Section. New York: Wiley, 1998.

[6] Marsaglia, George, and Wai
Wan Tsang. “A Simple Method for Generating Gamma Variables.” *ACM
Transactions on Mathematical Software* 26, no. 3 (September 1, 2000):
363–72. https://doi.org/10.1007/978-1-4613-8643-8.

`fitdist`

| `gamcdf`

| `gamfit`

| `gaminv`

| `gamlike`

| `GammaDistribution`

| `gampdf`

| `gamrnd`

| `gamstat`

| `makedist`

| `randg`