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# Cumulative Distribution Function

> [F][f-distribution] distribution [cumulative distribution function][cdf].

<section class="intro">

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

<!-- <equation class="equation" label="eq:f_cdf" align="center" raw="F(x; d_1,d_2)=I_{\frac{d_1 x}{d_1 x + d_2}}\left (\tfrac{d_1}{2}, \tfrac{d_2}{2} \right)" alt="Cumulative distribution function for an F distribution."> -->

<div class="equation" align="center" data-raw-text="F(x; d_1,d_2)=I_{\frac{d_1 x}{d_1 x + d_2}}\left (\tfrac{d_1}{2}, \tfrac{d_2}{2} \right)" data-equation="eq:f_cdf">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/stats/base/dists/f/cdf/docs/img/equation_f_cdf.svg" alt="Cumulative distribution function for an F distribution.">
    <br>
</div>

<!-- </equation> -->

where `d1` is the numerator degrees of freedom, `d2` is the denominator degrees of freedom and `I_{x}(a,b)` is the [lower regularized incomplete beta function][@stdlib/math/base/special/betainc].

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### cdf( x, d1, d2 )

Evaluates the [cumulative distribution function][cdf] (CDF) for a [F][f-distribution] distribution with parameters `d1` (numerator degrees of freedom) and `d2` (denominator degrees of freedom).

```javascript
var y = cdf( 2.0, 1.0, 1.0 );
// returns ~0.608

y = cdf( 2.0, 8.0, 4.0 );
// returns ~0.737

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

y = cdf( 0.0, 4.0, 4.0 );
// returns 0.0

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 `d1 <= 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 `d2 <= 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( d1, d2 )

Returns a function for evaluating the [cumulative distribution function][cdf] of a [F][f-distribution] distribution with parameters `d1` (numerator degrees of freedom) and `d2` (denominator degrees of freedom).

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

var y = mycdf( 10.0 );
// returns ~0.906

y = mycdf( 8.0 );
// returns ~0.884
```

</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/f/cdf' );

var d1;
var d2;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    d1 = randu() * 10.0;
    d2 = randu() * 10.0;
    y = cdf( x, d1, d2 );
    console.log( 'x: %d, d1: %d, d2: %d, F(x;d1,d2): %d', x.toFixed( 4 ), d1.toFixed( 4 ), d2.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

<section class="links">

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

[f-distribution]: https://en.wikipedia.org/wiki/F_distribution

[@stdlib/math/base/special/betainc]: https://github.com/stdlib-js/math-base-special-betainc

</section>

<!-- /.links -->
