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

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# Logarithm of 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 logarithm of 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@9dcb6eb6ab19f2ea81f3bcab5344b29961028a0c/lib/node_modules/@stdlib/stats/base/dists/gamma/logcdf/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-logcdf
```

</section>

<section class="usage">

## Usage

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

#### logcdf( x, alpha, beta )

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

```javascript
var y = logcdf( 2.0, 0.5, 1.0 );
// returns ~-0.047

y = logcdf( 0.1, 1.0, 1.0 );
// returns ~-2.352

y = logcdf( -1.0, 4.0, 2.0 );
// returns -Infinity
```

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

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

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

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

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

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

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

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

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

y = logcdf( 0.0, 0.0, 2.0 );
// returns 0.0
```

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

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

y = logcdf( 2.0, 1.0, -1.0 );
// returns NaN
```

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

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

```javascript
var mylogcdf = logcdf.factory( 3.0, 1.5 );

var y = mylogcdf( 1.0 );
// returns ~-1.655

y = mylogcdf( 4.0 );
// returns ~-0.064
```

* * *

### C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

#### Usage

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

#### stdlib_base_dists_gamma_logcdf( x, alpha, beta )

Evaluates the natural logarithm of the cumulative distribution function (CDF) for a gamma distribution with shape parameter `alpha` and rate parameter `beta`.

```c
double out = stdlib_base_dists_gamma_logcdf( 2.0, 0.5, 1.0 );
// returns ~-0.047

out = stdlib_base_dists_gamma_logcdf( 0.1, 1.0, 1.0 );
// returns ~-2.352
```

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_logcdf( const double x, const double alpha, const double beta );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

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

static double random_uniform( double a, double b ) {
    double r = ( (double)rand() / ( (double)RAND_MAX + 1.0 ) );
    return a + ( r * ( b - a ) );
}

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

    for ( i = 0; i < 25; i++ ) {
        x = random_uniform( 0.0, 2.0 );
        alpha = random_uniform( 1.0, 10.0 );
        beta = random_uniform( 1.0, 10.0 );
        y = stdlib_base_dists_gamma_logcdf( x, alpha, beta );
        printf( "x: %lf, α: %lf, β: %lf, ln(F(x;α,β)): %lf\n", x, alpha, beta, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.usage -->

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

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, ln(f(x;α,β)): %0.4f', x, alpha, beta, logcdf );
```

</section>

<!-- /.examples -->

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

<section class="related">

</section>

<!-- /.related -->

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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].

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---

## License

See [LICENSE][stdlib-license].


## Copyright

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

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

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

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

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