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# Cumulative Distribution Function

> [Arcsine][arcsine-distribution] distribution [cumulative distribution function][cdf].

<section class="intro">

The [cumulative distribution function][cdf] for an [arcsine][arcsine-distribution] random variable is

<!-- <equation class="equation" label="eq:arcsine_cdf" align="center" raw="F(x) = \frac{2}{\pi} \arcsin \left( \sqrt{\frac{x-a}{b-a}} \right)" alt="Cumulative distribution function for an arcsine distribution."> -->

<div class="equation" align="center" data-raw-text="F(x) = \frac{2}{\pi} \arcsin \left( \sqrt{\frac{x-a}{b-a}} \right)" data-equation="eq:arcsine_cdf">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/stats/base/dists/arcsine/cdf/docs/img/equation_arcsine_cdf.svg" alt="Cumulative distribution function for an arcsine distribution.">
    <br>
</div>

<!-- </equation> -->

where `a` is the minimum support and `b` is the maximum support. The parameters must satisfy `a < b`.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### cdf( x, a, b )

Evaluates the [cumulative distribution function][cdf] (CDF) for an [arcsine][arcsine-distribution] distribution with parameters `a` (minimum support) and `b` (maximum support).

```javascript
var y = cdf( 9.0, 0.0, 10.0 );
// returns ~0.795

y = cdf( 0.5, 0.0, 2.0 );
// returns ~0.333

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

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

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

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

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

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

If provided `a >= b`, the function returns `NaN`.

```javascript
var y = cdf( 1.0, 2.5, 2.0 );
// returns NaN
```

#### cdf.factory( a, b )

Returns a function for evaluating the [cumulative distribution function][cdf] of an [arcsine][arcsine-distribution] distribution with parameters `a` (minimum support) and `b` (maximum support).

```javascript
var mycdf = cdf.factory( 0.0, 10.0 );
var y = mycdf( 0.5 );
// returns ~0.144

y = mycdf( 8.0 );
// returns ~0.705
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random/base/randu' );
var cdf = require( '@stdlib/stats/base/dists/arcsine/cdf' );

var a;
var b;
var x;
var y;
var i;

for ( i = 0; i < 25; i++ ) {
    x = ( randu()*20.0 ) - 10.0;
    a = ( randu()*20.0 ) - 20.0;
    b = a + ( randu()*40.0 );
    y = cdf( x, a, b );
    console.log( 'x: %d, a: %d, b: %d, F(x;a,b): %d', x.toFixed( 4 ), a.toFixed( 4 ), b.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

<section class="links">

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

[arcsine-distribution]: https://en.wikipedia.org/wiki/Arcsine_distribution

</section>

<!-- /.links -->
