@stdlib/stats/base/dists/hypergeometric/pmf/README.md

Standard library statistical functions.

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# Probability Mass Function

> [Hypergeometric][hypergeometric-distribution] distribution [probability mass function][pmf] (PMF).

<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 [probability mass function][pmf] (PMF) for a [hypergeometric][hypergeometric-distribution] random variable is given by

<!-- <equation class="equation" label="eq:hypergeometric_pmf" align="center" raw="f(x;N,K,n)=P(X=x;N,K,n)=\begin{cases} {{{K \choose x} {N-K \choose {n-x}}}\over {{N} \choose n}} & \text{ for } x = 0,1,2,\ldots \\ 0 & \text{ otherwise} \end{cases}" alt="Probability mass function (PMF) for a hypergeometric distribution."> -->

<div class="equation" align="center" data-raw-text="f(x;N,K,n)=P(X=x;N,K,n)=\begin{cases} {{{K \choose x} {N-K \choose {n-x}}}\over {{N} \choose n}} &amp; \text{ for } x = 0,1,2,\ldots \\ 0 &amp; \text{ otherwise} \end{cases}" data-equation="eq:hypergeometric_pmf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/hypergeometric/pmf/docs/img/equation_hypergeometric_pmf.svg" alt="Probability mass function (PMF) for a hypergeometric distribution.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

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

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

```javascript
var y = pmf( 1.0, 8, 4, 2 );
// returns ~0.571

y = pmf( 2.0, 8, 4, 2 );
// returns ~0.214

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

y = pmf( 1.5, 8, 4, 2 );
// returns 0.0
```

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

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

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

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

y = pmf( 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 = pmf( 2.0, 10.5, 5, 2 );
// returns NaN

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

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

If the number of draws `n` exceeds population size `N`, the function returns `NaN`.

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

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

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

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

```javascript
var mypmf = pmf.factory( 30, 20, 5 );
var y = mypmf( 4.0 );
// returns ~0.34

y = mypmf( 1.0 );
// returns ~0.029
```

</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 pmf = require( '@stdlib/stats/base/dists/hypergeometric/pmf' );

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

for ( i = 0; i < 10; i++ ) {
    x = round( randu() * 5.0 );
    N = round( randu() * 20.0 );
    K = round( randu() * N );
    n = round( randu() * N );
    y = pmf( x, N, K, n );
    console.log( 'x: %d, N: %d, K: %d, n: %d, P(X=x;N,K,n): %d', x, N, K, n, 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>

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<section class="links">

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

[pmf]: https://en.wikipedia.org/wiki/Probability_mass_function

</section>

<!-- /.links -->