@stdlib/stats/base/dists/gamma/README.md

Standard library statistical functions.

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# Gamma

> Gamma distribution.

<section class="usage">

## Usage

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

#### gamma

Gamma distribution.

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

The namespace contains the following distribution functions:

<!-- <toc pattern="*+(cdf|pdf|mgf|quantile)*"> -->

<div class="namespace-toc">

-   <span class="signature">[`cdf( x, alpha, beta )`][@stdlib/stats/base/dists/gamma/cdf]</span><span class="delimiter">: </span><span class="description">gamma distribution cumulative distribution function.</span>
-   <span class="signature">[`logcdf( x, alpha, beta )`][@stdlib/stats/base/dists/gamma/logcdf]</span><span class="delimiter">: </span><span class="description">gamma distribution logarithm of cumulative distribution function (CDF).</span>
-   <span class="signature">[`logpdf( x, alpha, beta )`][@stdlib/stats/base/dists/gamma/logpdf]</span><span class="delimiter">: </span><span class="description">gamma distribution logarithm of probability density function (PDF).</span>
-   <span class="signature">[`mgf( t, alpha, beta )`][@stdlib/stats/base/dists/gamma/mgf]</span><span class="delimiter">: </span><span class="description">gamma distribution moment-generating function (MGF).</span>
-   <span class="signature">[`pdf( x, alpha, beta )`][@stdlib/stats/base/dists/gamma/pdf]</span><span class="delimiter">: </span><span class="description">gamma distribution probability density function (PDF).</span>
-   <span class="signature">[`quantile( p, alpha, beta )`][@stdlib/stats/base/dists/gamma/quantile]</span><span class="delimiter">: </span><span class="description">gamma distribution quantile function.</span>

</div>

<!-- </toc> -->

The namespace contains the following functions for calculating distribution properties:

<!-- <toc pattern="*+(entropy|kurtosis|mean|median|mode|skewness|stdev|variance)*"> -->

<div class="namespace-toc">

-   <span class="signature">[`entropy( alpha, beta )`][@stdlib/stats/base/dists/gamma/entropy]</span><span class="delimiter">: </span><span class="description">gamma distribution differential entropy.</span>
-   <span class="signature">[`kurtosis( alpha, beta )`][@stdlib/stats/base/dists/gamma/kurtosis]</span><span class="delimiter">: </span><span class="description">gamma distribution excess kurtosis.</span>
-   <span class="signature">[`mean( alpha, beta )`][@stdlib/stats/base/dists/gamma/mean]</span><span class="delimiter">: </span><span class="description">gamma distribution expected value.</span>
-   <span class="signature">[`mode( alpha, beta )`][@stdlib/stats/base/dists/gamma/mode]</span><span class="delimiter">: </span><span class="description">gamma distribution mode.</span>
-   <span class="signature">[`skewness( alpha, beta )`][@stdlib/stats/base/dists/gamma/skewness]</span><span class="delimiter">: </span><span class="description">gamma distribution skewness.</span>
-   <span class="signature">[`stdev( alpha, beta )`][@stdlib/stats/base/dists/gamma/stdev]</span><span class="delimiter">: </span><span class="description">gamma distribution standard deviation.</span>
-   <span class="signature">[`variance( alpha, beta )`][@stdlib/stats/base/dists/gamma/variance]</span><span class="delimiter">: </span><span class="description">gamma distribution variance.</span>

</div>

<!-- </toc> -->

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

<!-- <toc pattern="*ctor*"> -->

<div class="namespace-toc">

-   <span class="signature">[`Gamma( [alpha, beta] )`][@stdlib/stats/base/dists/gamma/ctor]</span><span class="delimiter">: </span><span class="description">gamma distribution constructor.</span>

</div>

<!-- </toc> -->

```javascript
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
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- TODO: better examples -->

<!-- eslint no-undef: "error" -->

```javascript
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/base/variance' );
var linspace = require( '@stdlib/array/base/linspace' );
var mean = require( '@stdlib/stats/base/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 );
```

</section>

<!-- /.examples -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

</section>

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<section class="links">

[gamma-distribution]: https://en.wikipedia.org/wiki/Gamma_distribution

<!-- <toc-links> -->

[@stdlib/stats/base/dists/gamma/ctor]: https://github.com/stdlib-js/stats/tree/main/base/dists/gamma/ctor

[@stdlib/stats/base/dists/gamma/entropy]: https://github.com/stdlib-js/stats/tree/main/base/dists/gamma/entropy

[@stdlib/stats/base/dists/gamma/kurtosis]: https://github.com/stdlib-js/stats/tree/main/base/dists/gamma/kurtosis

[@stdlib/stats/base/dists/gamma/mean]: https://github.com/stdlib-js/stats/tree/main/base/dists/gamma/mean

[@stdlib/stats/base/dists/gamma/mode]: https://github.com/stdlib-js/stats/tree/main/base/dists/gamma/mode

[@stdlib/stats/base/dists/gamma/skewness]: https://github.com/stdlib-js/stats/tree/main/base/dists/gamma/skewness

[@stdlib/stats/base/dists/gamma/stdev]: https://github.com/stdlib-js/stats/tree/main/base/dists/gamma/stdev

[@stdlib/stats/base/dists/gamma/variance]: https://github.com/stdlib-js/stats/tree/main/base/dists/gamma/variance

[@stdlib/stats/base/dists/gamma/cdf]: https://github.com/stdlib-js/stats/tree/main/base/dists/gamma/cdf

[@stdlib/stats/base/dists/gamma/logcdf]: https://github.com/stdlib-js/stats/tree/main/base/dists/gamma/logcdf

[@stdlib/stats/base/dists/gamma/logpdf]: https://github.com/stdlib-js/stats/tree/main/base/dists/gamma/logpdf

[@stdlib/stats/base/dists/gamma/mgf]: https://github.com/stdlib-js/stats/tree/main/base/dists/gamma/mgf

[@stdlib/stats/base/dists/gamma/pdf]: https://github.com/stdlib-js/stats/tree/main/base/dists/gamma/pdf

[@stdlib/stats/base/dists/gamma/quantile]: https://github.com/stdlib-js/stats/tree/main/base/dists/gamma/quantile

<!-- </toc-links> -->

</section>

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