@stdlib/stats/base/dists/beta/cdf/README.md

Standard library statistical functions.

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# Cumulative Distribution Function

> [Beta][beta-distribution] distribution [cumulative distribution function][cdf].

<section class="intro">

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

<!-- <equation class="equation" label="eq:beta_cdf" align="center" raw="F(x;\alpha,\beta) = \frac{\operatorname{Beta}(x;\alpha,\beta)}{\operatorname{Beta}(\alpha,\beta)}" alt="Cumulative distribution function for a beta distribution."> -->

<div class="equation" align="center" data-raw-text="F(x;\alpha,\beta) = \frac{\operatorname{Beta}(x;\alpha,\beta)}{\operatorname{Beta}(\alpha,\beta)}" data-equation="eq:beta_cdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/beta/cdf/docs/img/equation_beta_cdf.svg" alt="Cumulative distribution function for a beta distribution.">
    <br>
</div>

<!-- </equation> -->

where `alpha > 0` is the first shape parameter and `beta > 0` is the second shape parameter. In the definition, `Beta( x; a, b )` denotes the lower incomplete beta function and `Beta( a, b )` the [beta function][beta-function].

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### cdf( x, alpha, beta )

Evaluates the [cumulative distribution function][cdf] (CDF) for a [beta][beta-distribution] distribution with parameters `alpha` (first shape parameter) and `beta` (second shape parameter).

```javascript
var y = cdf( 0.5, 1.0, 1.0 );
// returns 0.5

y = cdf( 0.5, 2.0, 4.0 );
// returns ~0.813

y = cdf( 0.2, 2.0, 2.0 );
// returns ~0.104

y = cdf( 0.8, 4.0, 4.0 );
// returns ~0.967

y = cdf( -0.5, 4.0, 2.0 );
// returns 0.0

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

y = cdf( 1.5, 4.0, 2.0 );
// returns 1.0

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

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

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

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

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

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

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

y = cdf( 2.0, 0.0, 0.5 );
// returns NaN
```

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

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

y = cdf( 2.0, 0.5, 0.0 );
// returns NaN
```

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

Returns a function for evaluating the [cumulative distribution function][cdf] for a [beta][beta-distribution] distribution with parameters `alpha` (first shape parameter) and `beta` (second shape parameter).

```javascript
var mycdf = cdf.factory( 0.5, 0.5 );

var y = mycdf( 0.8 );
// returns ~0.705

y = mycdf( 0.3 );
// returns ~0.369
```

</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/beta/cdf' );

var alpha;
var beta;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu();
    alpha = ( randu()*5.0 ) + EPS;
    beta = ( randu()*5.0 ) + EPS;
    y = cdf( x, alpha, beta );
    console.log( 'x: %d, α: %d, β: %d, F(x;α,β): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.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">

[beta-distribution]: https://en.wikipedia.org/wiki/Beta_distribution

[beta-function]: https://en.wikipedia.org/wiki/Beta_function

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

</section>

<!-- /.links -->