npm package: @stdlib/stats-base-dists-gamma-cdf
Description: Gamma distribution cumulative distribution function (CDF).
Author: The Stdlib Authors
Latest version: 0.3.1

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# Cumulative Distribution Function

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> [Gamma][gamma-distribution] distribution [cumulative distribution function][cdf].

<section class="intro">

The [cumulative distribution function][cdf] for a [gamma][gamma-distribution] random variable is

<!-- <equation class="equation" label="eq:gamma_cdf" align="center" raw="F(x;\alpha,\beta) = \int_0^x f(u;\alpha,\beta)\,du= \frac{\gamma(\alpha, \beta x)}{\Gamma(\alpha)}" alt="Cumulative distribution function for a Gamma distribution."> -->

<div class="equation" align="center" data-raw-text="F(x;\alpha,\beta) = \int_0^x f(u;\alpha,\beta)\,du= \frac{\gamma(\alpha, \beta x)}{\Gamma(\alpha)}" data-equation="eq:gamma_cdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/gamma/cdf/docs/img/equation_gamma_cdf.svg" alt="Cumulative distribution function for a Gamma distribution.">
    <br>
</div>

<!-- </equation> -->

where `alpha` is the shape parameter and `beta` is the rate parameter of the distribution. `gamma` is the lower [incomplete gamma function][@stdlib/math/base/special/gammainc].

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-gamma-cdf
```

</section>

<section class="usage">

## Usage

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

#### cdf( x, alpha, beta )

Evaluates the [cumulative distribution function][cdf] (CDF) for a [gamma][gamma-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (rate parameter).

```javascript
var y = cdf( 2.0, 1.0, 1.0 );
// returns ~0.865

y = cdf( 2.0, 3.0, 1.0 );
// returns ~0.323

y = cdf( -1.0, 2.0, 2.0 );
// returns 0.0

y = cdf( -Infinity, 4.0, 2.0 );
// returns 0.0

y = cdf( +Infinity, 4.0, 2.0 );
// returns 1.0
```

If provided `NaN` as any argument, the function returns `NaN`.

```javascript
var y = cdf( NaN, 1.0, 1.0 );
// returns NaN

y = cdf( 0.0, NaN, 1.0 );
// returns NaN

y = cdf( 0.0, 1.0, NaN );
// returns NaN
```

If provided `alpha < 0`, the function returns `NaN`.

```javascript
var y = cdf( 2.0, -1.0, 0.5 );
// returns NaN
```

If provided `alpha = 0`, the function evaluates the [CDF][cdf] of a [degenerate distribution][degenerate-distribution] centered at `0`.

```javascript
var y = cdf( 2.0, 0.0, 2.0 );
// returns 1.0

y = cdf( -2.0, 0.0, 2.0 );
// returns 0.0

y = cdf( 0.0, 0.0, 2.0 );
// returns 1.0
```

If provided `beta <= 0`, the function returns `NaN`.

```javascript
var y = cdf( 2.0, 0.5, -1.0 );
// returns NaN
```

#### cdf.factory( alpha, beta )

Returns a function for evaluating the [cumulative distribution function][cdf] for a [gamma][gamma-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (rate parameter).

```javascript
var mycdf = cdf.factory( 0.5, 0.1 );

var y = mycdf( 12.0 );
// returns ~0.879

y = mycdf( 8.0 );
// returns ~0.794
```

</section>

<!-- /.usage -->

* * *

<section class="c">

## C APIs

<section class="intro">

</section>

<!-- /.intro -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/gamma/cdf.h"
```

#### stdlib_base_dists_gamma_cdf( x, alpha, beta )

Evaluates the [cumulative distribution function][cdf] (CDF) for a [gamma][gamma-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (rate parameter).

```c
double out = stdlib_base_dists_gamma_cdf( 2.0, 1.0, 1.0 );
// returns ~0.865

out = stdlib_base_dists_gamma_cdf( 2.0, 3.0, 1.0 );
// returns ~0.323
```

The function accepts the following arguments:

-   **x**: `[in] double` input value.
-   **alpha**: `[in] double` shape parameter.
-   **beta**: `[in] double` rate parameter.

```c
double stdlib_base_dists_gamma_cdf( const double x, const double alpha, const double beta );
```

</section>

<!-- /.usage -->

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/gamma/cdf.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( double min, double max ) {
    double scale = rand() / (double) RAND_MAX;
    return min + ( scale * ( max - min ) );
}

int main( void ) {
    double x;
    double alpha;
    double beta;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        x = random_uniform( 0.0, 3.0 );
        alpha = random_uniform( 0.1, 5.0 );
        beta = random_uniform( 0.1, 5.0 );
        y = stdlib_base_dists_gamma_cdf( x, alpha, beta );
        printf( "x: %lf, α: %lf, β: %lf, F(x;α,β): %lf\n", x, alpha, beta, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

* * *

<section class="examples">

## Examples

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

```javascript
var uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var cdf = require( '@stdlib/stats-base-dists-gamma-cdf' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 10, 0.0, 3.0, opts );
var alpha = uniform( 10, 0.0, 5.0, opts );
var beta = uniform( 10, 0.0, 5.0, opts );

logEachMap( 'x: %0.4f, α: %0.4f, β: %0.4f, F(x;α,β): %0.4f', x, alpha, beta, cdf );
```

</section>

<!-- /.examples -->

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<section class="related">

</section>

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<section class="main-repo" >

* * *

## Notice

This package is part of [stdlib][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][stdlib], see the main project [repository][stdlib].

#### Community

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---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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[cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function

[degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution

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

[@stdlib/math/base/special/gammainc]: https://www.npmjs.com/package/@stdlib/math-base-special-gammainc

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