npm package: @stdlib/stats-base-dists-cauchy-cdf
Description: Cauchy 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] -->

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

<section class="intro">

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

<!-- <equation class="equation" label="eq:cauchy_cdf" align="center" raw="F(x; x_0,\gamma)=\frac{1}{\pi} \operatorname{arctan} \left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}" alt="Cumulative distribution function for a Cauchy distribution."> -->

<div class="equation" align="center" data-raw-text="F(x; x_0,\gamma)=\frac{1}{\pi} \operatorname{arctan} \left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}" data-equation="eq:cauchy_cdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/docs/img/equation_cauchy_cdf.svg" alt="Cumulative distribution function for a Cauchy distribution.">
    <br>
</div>

<!-- </equation> -->

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

</section>

<!-- /.intro -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## 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
```

</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 EPS = require( '@stdlib/constants-float64-eps' );
var cdf = require( '@stdlib/stats-base-dists-cauchy-cdf' );

var opts = {
    'dtype': 'float64'
};
var gamma = uniform( 10, EPS, 10.0, opts );
var x0 = uniform( 10, 0.0, 10.0, opts );
var x = uniform( 10, 0.0, 10.0, opts );

logEachMap( 'x: %0.4f, x0: %0.4f, γ: %0.4f, F(x;x0,γ): %0.4f', x, x0, gamma, cdf );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## 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/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 );
```

</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/cauchy/cdf.h"
#include <stdlib.h>
#include <stdio.h>

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 );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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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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[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

</section>

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