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# Cumulative Distribution Function

> [Hypergeometric][hypergeometric-distribution] distribution [cumulative distribution function][cdf].

<section class="intro">

Imagine a scenario with a population of size `N`, of which a subpopulation of size `K` can be considered successes. We draw `n` observations from the total population. Defining the random variable `X` as the number of successes in the `n` draws, `X` is said to follow a [hypergeometric distribution][hypergeometric-distribution]. The [cumulative distribution function][cdf] for a [hypergeometric][hypergeometric-distribution] random variable is

<!-- <equation class="equation" label="eq:hypergeometric_cdf" align="center" raw="F(x;N,K,n) =\sum_{i=0}^{\lfloor x \rfloor} \frac{{K \choose i}{N-K \choose n-i}}{{N \choose n}}" alt="Cumulative distribution function for a hypergeometric distribution."> -->

<div class="equation" align="center" data-raw-text="F(x;N,K,n) =\sum_{i=0}^{\lfloor x \rfloor} \frac{{K \choose i}{N-K \choose n-i}}{{N \choose n}}" data-equation="eq:hypergeometric_cdf">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/stats/base/dists/hypergeometric/cdf/docs/img/equation_hypergeometric_cdf.svg" alt="Cumulative distribution function for a hypergeometric distribution.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### cdf( x, N, K, n )

Evaluates the [cumulative distribution function][cdf] for a [hypergeometric][hypergeometric-distribution] distribution with parameters `N` (population size), `K` (subpopulation size), and `n` (number of draws).

```javascript
var y = cdf( 1.0, 8, 4, 2 );
// returns ~0.786

y = cdf( 1.5, 8, 4, 2 );
// returns ~0.786

y = cdf( 2.0, 8, 4, 2 );
// returns 1.0

y = cdf( 0.0, 8, 4, 2);
// returns ~0.214
```

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

```javascript
var y = cdf( NaN, 10, 5, 2 );
// returns NaN

y = cdf( 0.0, NaN, 5, 2 );
// returns NaN

y = cdf( 0.0, 10, NaN, 2 );
// returns NaN

y = cdf( 0.0, 10, 5, NaN );
// returns NaN
```

If provided a population size `N`, subpopulation size `K` or draws `n` which is not a nonnegative integer, the function returns `NaN`.

```javascript
var y = cdf( 2.0, 10.5, 5, 2 );
// returns NaN

y = cdf( 2.0, 10, 1.5, 2 );
// returns NaN

y = cdf( 2.0, 10, 5, -2.0 );
// returns NaN
```

If the number of draws `n` or subpopulation size `K` exceed the population size `N`, the function returns `NaN`.

```javascript
var y = cdf( 2.0, 10, 5, 12 );
// returns NaN

y = cdf( 2.0, 8, 3, 9 );
// returns NaN
```

#### cdf.factory( N, K, n )

Returns a function for evaluating the [cumulative distribution function][cdf] of a [hypergeometric ][hypergeometric-distribution] distribution with parameters `N` (population size), `K` (subpopulation size), and `n` (number of draws).

```javascript
var mycdf = cdf.factory( 30, 20, 5 );
var y = mycdf( 4.0 );
// returns ~0.891

y = mycdf( 1.0 );
// returns ~0.031
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var cdf = require( '@stdlib/stats/base/dists/hypergeometric/cdf' );

var i;
var N;
var K;
var n;
var x;
var y;

for ( i = 0; i < 10; i++ ) {
    N = round( randu() * 20 );
    K = round( randu() * N );
    n = round( randu() * K );
    x = round( randu() * K );
    y = cdf( x, N, K, n );
    console.log( 'x: %d, N: %d, K: %d, n: %d, F(x;N,K,n): %d', x.toFixed( 4 ), N, K, n, y.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

<section class="links">

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

[hypergeometric-distribution]: https://en.wikipedia.org/wiki/hypergeometric_distribution

</section>

<!-- /.links -->
