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# Cumulative Distribution Function

> [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="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 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 ) );
}
```

</section>

<!-- /.examples -->

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

<section class="related">

</section>

<!-- /.related -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="links">

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

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

</section>

<!-- /.links -->
