Gamma

Gamma distribution.

Installation

npm install @stdlib/stats-base-dists-gamma

Usage

var gamma = require( '@stdlib/stats-base-dists-gamma' );

gamma

Gamma distribution.

var dist = gamma;
// returns {...}

The namespace contains the following distribution functions:

• cdf( x, alpha, beta ): gamma distribution cumulative distribution function.
• logcdf( x, alpha, beta ): gamma distribution logarithm of cumulative distribution function (CDF).
• logpdf( x, alpha, beta ): gamma distribution logarithm of probability density function (PDF).
• mgf( t, alpha, beta ): gamma distribution moment-generating function (MGF).
• pdf( x, alpha, beta ): gamma distribution probability density function (PDF).
• quantile( p, alpha, beta ): gamma distribution quantile function.

The namespace contains the following functions for calculating distribution properties:

• entropy( alpha, beta ): gamma distribution differential entropy.
• kurtosis( alpha, beta ): gamma distribution excess kurtosis.
• mean( alpha, beta ): gamma distribution expected value.
• mode( alpha, beta ): gamma distribution mode.
• skewness( alpha, beta ): gamma distribution skewness.
• stdev( alpha, beta ): gamma distribution standard deviation.
• variance( alpha, beta ): gamma distribution variance.

The namespace contains a constructor function for creating a gamma distribution object.

• Gamma( [alpha, beta] ): gamma distribution constructor.

var Gamma = require( '@stdlib/stats-base-dists-gamma' ).Gamma;

var dist = new Gamma( 2.0, 4.0 );

var y = dist.cdf( 0.5 );
// returns ~0.594

Examples

var gammaRandomFactory = require( '@stdlib/random-base-gamma' ).factory;
var filledarrayby = require( '@stdlib/array-filled-by' );
var Float64Array = require( '@stdlib/array-float64' );
var variance = require( '@stdlib/stats-strided-variance' );
var linspace = require( '@stdlib/array-base-linspace' );
var mean = require( '@stdlib/stats-strided-mean' );
var abs = require( '@stdlib/math-base-special-abs' );
var gamma = require( '@stdlib/stats-base-dists-gamma' );

// Define the shape and scale parameters:
var alpha = 3.0; // shape parameter (α)
var beta = 2.0;  // scale parameter (β)

// Generate an array of x values:
var x = linspace( 0.0, 20.0, 100 );

// Compute the PDF for each x:
var gammaPDF = gamma.pdf.factory( alpha, beta );
var pdf = filledarrayby( x.length, 'float64', gammaPDF );

// Compute the CDF for each x:
var gammaCDF = gamma.cdf.factory( alpha, beta );
var cdf = filledarrayby( x.length, 'float64', gammaCDF );

// Output the PDF and CDF values:
console.log( 'x values: %s', x );
console.log( 'PDF values: %s', pdf );
console.log( 'CDF values: %s', cdf );

// Compute statistical properties:
var theoreticalMean = gamma.mean( alpha, beta );
var theoreticalVariance = gamma.variance( alpha, beta );
var theoreticalSkewness = gamma.skewness( alpha, beta );
var theoreticalKurtosis = gamma.kurtosis( alpha, beta );

console.log( 'Theoretical Mean: %s', theoreticalMean );
console.log( 'Theoretical Variance: %s', theoreticalVariance );
console.log( 'Skewness: %s', theoreticalSkewness );
console.log( 'Kurtosis: %s', theoreticalKurtosis );

// Generate random samples from the gamma distribution:
var rgamma = gammaRandomFactory( alpha, beta );
var n = 300;
var samples = filledarrayby( n, 'float64', rgamma );

// Compute sample mean and variance:
var sampleMean = mean( n, samples, 1 );
var sampleVariance = variance( n, 1, samples, 1 );

console.log( 'Sample Mean: %s', sampleMean );
console.log( 'Sample Variance: %s', sampleVariance );

// Compare sample statistics to theoretical values:
console.log( 'Difference in Mean: %s', abs( theoreticalMean - sampleMean ) );
console.log( 'Difference in Variance: %s', abs( theoreticalVariance - sampleVariance ) );

// Demonstrate that the sum of `k` gamma variables is a gamma-distributed sum of `k` gamma(α, β) variables with same β is `gamma(k*α, β)`:
var k = 5;
var sumSamples = new Float64Array( n );

var sum;
var i;
var j;
for ( i = 0; i < sumSamples.length; i++ ) {
    sum = 0.0;
    for ( j = 0; j < k; j++ ) {
        sum += rgamma();
    }
    sumSamples[ i ] = sum;
}

// Theoretical parameters for the sum:
var sumAlpha = k * alpha;
var sumMean = gamma.mean( sumAlpha, beta );
var sumVariance = gamma.variance( sumAlpha, beta );

console.log( 'Sum Theoretical Mean: %s', sumMean );
console.log( 'Sum Theoretical Variance: %s', sumVariance );

// Compute sample mean and variance for the sum:
var sumSampleMean = mean( sumSamples.length, sumSamples, 1 );
var sumSampleVariance = variance( sumSamples.length, 1, sumSamples, 1 );

console.log( 'Sum Sample Mean: %s', sumSampleMean );
console.log( 'Sum Sample Variance: %s', sumSampleVariance );

Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

License

See LICENSE.

Copyright

Copyright © 2016-2026. The Stdlib Authors.