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

Standard library statistical functions.

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# Quantile Function

> [Gamma][gamma-distribution] distribution [quantile function][quantile-function].

<section class="intro">

The [quantile function][quantile-function] for a [gamma][gamma-distribution] random variable is

<!-- <equation class="equation" label="eq:gamma_quantile_function" align="center" raw="Q(p;\alpha,\beta) = \frac{1}{\beta} P^{-1}\left( p, \alpha \right )" data-equation="eq:quantile_function" alt="Quantile function for a Gamma distribution."> -->

<div class="equation" align="center" data-raw-text="Q(p;\alpha,\beta) = \frac{1}{\beta} P^{-1}\left( p, \alpha \right )" data-equation="eq:gamma_quantile_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/gamma/quantile/docs/img/equation_gamma_quantile_function.svg" alt="Quantile function for a Gamma distribution.">
    <br>
</div>

<!-- </equation> -->

for `0 <= p < 1`, where `alpha` is the shape parameter and `beta` is the rate parameter of the distribution. `P^{-1}` is the inverse of the lower regularized incomplete gamma function.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

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

Evaluates the [quantile function][quantile-function] for a [gamma][gamma-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (rate parameter).

```javascript
var y = quantile( 0.8, 2.0, 1.0 );
// returns ~2.994

y = quantile( 0.5, 4.0, 2.0 );
// returns ~1.836
```

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( 0.0, NaN, 1.0 );
// returns NaN

y = quantile( 0.0, 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
```

If provided `alpha = 0`, the function evaluates the [quantile function][quantile-function] of a [degenerate distribution][degenerate-distribution] centered at `0`.

```javascript
var y = quantile( 0.3, 0.0, 2.0 );
// returns 0.0

y = quantile( 0.9, 0.0, 2.0 );
// returns 0.0
```

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

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

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

Returns a function for evaluating the [quantile function][quantile-function] of a [gamma][gamma-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (rate parameter).

```javascript
var myquantile = quantile.factory( 2.0, 2.0 );
var y = myquantile( 0.8 );
// returns ~1.497

y = myquantile( 0.4 );
// returns ~0.688
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

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

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

for ( i = 0; i < 20; i++ ) {
    p = randu();
    alpha = randu() * 5.0;
    beta = randu() * 5.0;
    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">

[degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution

[gamma-distribution]: https://en.wikipedia.org/wiki/Gamma_distribution

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

</section>

<!-- /.links -->