@stdlib/stats/base/dists/betaprime/quantile/README.md

Standard library statistical functions.

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# Quantile Function

> [Beta prime][betaprime-distribution] distribution [quantile function][quantile-function].

<section class="intro">

The [quantile function][quantile-function] for a [beta prime][betaprime-distribution] random variable with first shape parameter `α > 0` and second shape parameter `β > 0` is

<!-- <equation class="equation" label="eq:betaprime_quantile_function" align="center" raw="Q(p;\alpha,\beta)\,= \frac{G^{-1}(p)}{1-G^{-1}(p)}" alt="Quantile function for a beta prime distribution."> -->

<div class="equation" align="center" data-raw-text="Q(p;\alpha,\beta)\,= \frac{G^{-1}(p)}{1-G^{-1}(p)}" data-equation="eq:betaprime_quantile_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/betaprime/quantile/docs/img/equation_betaprime_quantile_function.svg" alt="Quantile function for a beta prime distribution.">
    <br>
</div>

<!-- </equation> -->

for `0 <= p <= 1`, where `G^-1` denotes the quantile function of a [beta][beta-distribution] random variable with parameters `α` and `β`.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var quantile = require( '@stdlib/stats/base/dists/betaprime/quantile' );
```

#### quantile( p, alpha, beta )

Evaluates the [quantile function][quantile-function] for a [beta prime][betaprime-distribution] distribution with parameters `alpha` (first shape parameter) and `beta` (second shape parameter).

```javascript
var y = quantile( 0.8, 2.0, 1.0 );
// returns ~8.472

y = quantile( 0.5, 4.0, 2.0 );
// returns ~2.187
```

If provided a probability `p` outside the interval `[0,1]`, the function returns `NaN`.

```javascript
var y = quantile( 1.9, 1.0, 1.0 );
// returns NaN

y = quantile( -0.1, 1.0, 1.0 );
// returns NaN
```

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

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

y = quantile( 0.5, NaN, 1.0 );
// returns NaN

y = quantile( 0.5, 1.0, NaN );
// returns NaN
```

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

```javascript
var y = quantile( 0.4, -1.0, 1.0 );
// returns NaN

y = quantile( 0.4, 0.0, 1.0 );
// returns NaN
```

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

```javascript
var y = quantile( 0.4, 1.0, -1.0 );
// returns NaN

y = quantile( 0.4, 1.0, 0.0 );
// returns NaN
```

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

Returns a function for evaluating the [quantile function][quantile-function] of a [beta prime][betaprime-distribution] distribution with parameters `alpha` (first shape parameter) and `beta` (second shape parameter).

```javascript
var myQuantile = quantile.factory( 2.0, 2.0 );

var y = myQuantile( 0.8 );
// returns ~2.483

y = myQuantile( 0.4 );
// returns ~0.763
```

</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 quantile = require( '@stdlib/stats/base/dists/betaprime/quantile' );

var alpha;
var beta;
var p;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    p = randu();
    alpha = ( randu()*5.0 ) + EPS;
    beta = ( randu()*5.0 ) + EPS;
    y = quantile( p, alpha, beta );
    console.log( 'p: %d, α: %d, β: %d, Q(p;α,β): %d', p.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

[betaprime-distribution]: https://en.wikipedia.org/wiki/Beta_prime_distribution

[quantile-function]: https://en.wikipedia.org/wiki/Quantile_function

</section>

<!-- /.links -->