https://github.com/stdlib-js/stats-base-dists-cauchy-cdf

Cauchy distribution cumulative distribution function (CDF).
https://github.com/stdlib-js/stats-base-dists-cauchy-cdf

cauchy cdf continuous cumulative cumulative-distribution dist distribution distribution-function javascript node node-js nodejs prob probability statistics stats stdlib univariate

Last synced: about 1 month ago
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Cauchy distribution cumulative distribution function (CDF).

Host: GitHub
URL: https://github.com/stdlib-js/stats-base-dists-cauchy-cdf
Owner: stdlib-js
License: apache-2.0
Created: 2021-06-15T17:44:26.000Z (almost 4 years ago)
Default Branch: main
Last Pushed: 2025-04-01T06:51:13.000Z (about 2 months ago)
Last Synced: 2025-04-13T00:36:46.316Z (about 1 month ago)
Topics: cauchy, cdf, continuous, cumulative, cumulative-distribution, dist, distribution, distribution-function, javascript, node, node-js, nodejs, prob, probability, statistics, stats, stdlib, univariate
Language: Python
Homepage: https://github.com/stdlib-js/stdlib
Size: 1.05 MB
Stars: 2
Watchers: 3
Forks: 0
Open Issues: 0
Metadata Files:
- Readme: README.md
- Changelog: CHANGELOG.md
- Contributing: CONTRIBUTING.md
- License: LICENSE
- Code of conduct: CODE_OF_CONDUCT.md
- Citation: CITATION.cff
- Security: SECURITY.md

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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] 

> [Cauchy][cauchy-distribution] distribution [cumulative distribution function][cdf].

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

```math

F(x; x_0,\gamma)=\frac{1}{\pi} \mathop{\mathrm{arctan}} \left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}

```

where `x0` is the location parameter and `gamma > 0` is the scale parameter.

## Installation

```bash

npm install @stdlib/stats-base-dists-cauchy-cdf

```

Alternatively,

-   To load the package in a website via a `script` tag without installation and bundlers, use the [ES Module][es-module] available on the [`esm`][esm-url] branch (see [README][esm-readme]).

-   If you are using Deno, visit the [`deno`][deno-url] branch (see [README][deno-readme] for usage intructions).

-   For use in Observable, or in browser/node environments, use the [Universal Module Definition (UMD)][umd] build available on the [`umd`][umd-url] branch (see [README][umd-readme]).

The [branches.md][branches-url] file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

## Usage

```javascript

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

```

#### cdf( x, x0, gamma )

Evaluates the [cumulative distribution function][cdf] (CDF) for a [Cauchy][cauchy-distribution] distribution with parameters `x0` (location parameter) and `gamma > 0` (scale parameter).

```javascript

var y = cdf( 4.0, 0.0, 2.0 );

// returns ~0.852

y = cdf( 1.0, 0.0, 2.0 );

// returns ~0.648

y = cdf( 1.0, 3.0, 2.0 );

// returns 0.25

```

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

```javascript

var y = cdf( NaN, 0.0, 2.0 );

// returns NaN

y = cdf( 1.0, 2.0, NaN );

// returns NaN

y = cdf( 1.0, NaN, 3.0 );

// returns NaN

```

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

```javascript

var y = cdf( 2.0, 0.0, -1.0 );

// returns NaN

y = cdf( 2.0, 0.0, 0.0 );

// returns NaN

```

#### cdf.factory( x0, gamma )

Returns a function for evaluating the [cumulative distribution function][cdf] of a [Cauchy][cauchy-distribution] distribution with parameters `x0` (location parameter) and `gamma > 0` (scale parameter).

```javascript

var mycdf = cdf.factory( 10.0, 2.0 );

var y = mycdf( 10.0 );

// returns 0.5

y = mycdf( 12.0 );

// returns 0.75

```

## Examples

```javascript

var randu = require( '@stdlib/random-base-randu' );

var EPS = require( '@stdlib/constants-float64-eps' );

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

var gamma;

var x0;

var x;

var y;

var i;

for ( i = 0; i < 10; i++ ) {

    x = randu() * 10.0;

    x0 = randu() * 10.0;

    gamma = ( randu()*10.0 ) + EPS;

    y = cdf( x, x0, gamma );

    console.log( 'x: %d, x0: %d, γ: %d, F(x;x0,γ): %d', x.toFixed( 4 ), x0.toFixed( 4 ), gamma.toFixed( 4 ), y.toFixed( 4 ) );

}

```

* * *

## C APIs

### Usage

```c

#include "stdlib/stats/base/dists/cauchy/cdf.h"

```

#### stdlib_base_dists_cauchy_cdf( x, x0, gamma )

Evaluates the [cumulative distribution function][cdf] (CDF) for a [Cauchy][cauchy-distribution] distribution with parameters `x0` (location parameter) and `gamma > 0` (scale parameter).

```c

double out = stdlib_base_dists_cauchy_cdf( 4.0, 0.0, 2.0 );

// returns ~0.852

```

The function accepts the following arguments:

-   **x**: `[in] double` input value.

-   **x0**: `[in] double` location parameter.

-   **gamma**: `[in] double` scale parameter.

```c

double stdlib_base_dists_cauchy_cdf( const double x, const double x0, const double gamma );

```

### Examples

```c

#include "stdlib/stats/base/dists/cauchy/cdf.h"

#include 

#include 

static double random_uniform( const double min, const double max ) {

    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );

    return min + ( v*(max-min) );

}

int main( void ) {

    double gamma;

    double x0;

    double y;

    double x;

    int i;

    for ( i = 0; i < 25; i++ ) {

        x = random_uniform( 0.0, 10.0 );

        x0 = random_uniform( 0.0, 10.0 );

        gamma = random_uniform( 0.0, 10.0 );

        y = stdlib_base_dists_cauchy_cdf( x, x0, gamma );

        printf( "x: %lf, k: %lf, γ: %lf, F(x;x0,γ): %lf\n", x, x0, gamma, y );

    }

}

```

* * *

## 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

[![Chat][chat-image]][chat-url]

---

## License

See [LICENSE][stdlib-license].

## Copyright

Copyright © 2016-2025. The Stdlib [Authors][stdlib-authors].

[npm-image]: http://img.shields.io/npm/v/@stdlib/stats-base-dists-cauchy-cdf.svg

[npm-url]: https://npmjs.org/package/@stdlib/stats-base-dists-cauchy-cdf

[test-image]: https://github.com/stdlib-js/stats-base-dists-cauchy-cdf/actions/workflows/test.yml/badge.svg?branch=main

[test-url]: https://github.com/stdlib-js/stats-base-dists-cauchy-cdf/actions/workflows/test.yml?query=branch:main

[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/stats-base-dists-cauchy-cdf/main.svg

[coverage-url]: https://codecov.io/github/stdlib-js/stats-base-dists-cauchy-cdf?branch=main

[chat-image]: https://img.shields.io/gitter/room/stdlib-js/stdlib.svg

[chat-url]: https://app.gitter.im/#/room/#stdlib-js_stdlib:gitter.im

[stdlib]: https://github.com/stdlib-js/stdlib

[stdlib-authors]: https://github.com/stdlib-js/stdlib/graphs/contributors

[umd]: https://github.com/umdjs/umd

[es-module]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Modules

[deno-url]: https://github.com/stdlib-js/stats-base-dists-cauchy-cdf/tree/deno

[deno-readme]: https://github.com/stdlib-js/stats-base-dists-cauchy-cdf/blob/deno/README.md

[umd-url]: https://github.com/stdlib-js/stats-base-dists-cauchy-cdf/tree/umd

[umd-readme]: https://github.com/stdlib-js/stats-base-dists-cauchy-cdf/blob/umd/README.md

[esm-url]: https://github.com/stdlib-js/stats-base-dists-cauchy-cdf/tree/esm

[esm-readme]: https://github.com/stdlib-js/stats-base-dists-cauchy-cdf/blob/esm/README.md

[branches-url]: https://github.com/stdlib-js/stats-base-dists-cauchy-cdf/blob/main/branches.md

[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-base-dists-cauchy-cdf/main/LICENSE

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

[cauchy-distribution]: https://en.wikipedia.org/wiki/Cauchy_distribution

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