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# Quantile Function
> [Triangular][triangular-distribution] distribution [quantile function][quantile-function].
<section class="intro">
The [quantile function][quantile-function] for a [Triangular][triangular-distribution] random variable is
<!-- <equation class="equation" label="eq:triangular_quantile_function" align="center" raw="Q(p;a,b,c) = \begin{cases} a + \sqrt{(b-a)(c-a)p} & \text{ for } 0 \le p \le F(c) \\ b - \sqrt{(b-a)(b-c)(1-p)} & \text{ for } F(c) \le p \le 1 \end{cases}" alt="Quantile function for a triangular distribution."> -->
<div class="equation" align="center" data-raw-text="Q(p;a,b,c) = \begin{cases} a + \sqrt{(b-a)(c-a)p} & \text{ for } 0 \le p \le F(c) \\ b - \sqrt{(b-a)(b-c)(1-p)} & \text{ for } F(c) \le p \le 1 \end{cases}" data-equation="eq:triangular_quantile_function">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/triangular/quantile/docs/img/equation_triangular_quantile_function.svg" alt="Quantile function for a triangular distribution.">
<br>
</div>
<!-- </equation> -->
where `a` is the lower limit, `b` is the upper limit and `c` is the mode.
</section>
<!-- /.intro -->
<section class="usage">
## Usage
```javascript
var quantile = require( '@stdlib/stats/base/dists/triangular/quantile' );
```
#### quantile( p, a, b, c )
Evaluates the [quantile function][quantile-function] for a [triangular][triangular-distribution] distribution with parameters `a` (lower limit), `b` (upper limit) and `c` (mode).
```javascript
var y = quantile( 0.9, -1.0, 1.0, 0.0 );
// returns ~0.553
y = quantile( 0.1, -1.0, 1.0, 0.5 );
// returns ~-0.452
y = quantile( 0.1, -20.0, 0.0, -2.0 );
// returns -14.0
y = quantile( 0.8, 0.0, 20.0, 0.0 );
// returns ~11.056
```
If provided a probability `p` outside the interval `[0,1]`, the function returns `NaN`.
```javascript
var y = quantile( 1.9, 0.0, 1.0, 0.5 );
// returns NaN
y = quantile( -0.1, 0.0, 1.0, 0.5 );
// returns NaN
```
If provided `NaN` as any argument, the function returns `NaN`.
```javascript
var y = quantile( NaN, 0.0, 1.0, 0.5 );
// returns NaN
y = quantile( 0.1, NaN, 1.0, 0.5 );
// returns NaN
y = quantile( 0.1, 0.0, NaN, 0.5 );
// returns NaN
y = quantile( 0.1, 0.0, 1.0, NaN );
// returns NaN
```
If provided parameters not satisfying `a <= c <= b`, the function returns `NaN`.
```javascript
var y = quantile( 0.1, 1.0, 0.0, 1.5 );
// returns NaN
y = quantile( 0.1, 1.0, 0.0, -1.0 );
// returns NaN
y = quantile( 0.1, 0.0, -1.0, 0.5 );
// returns NaN
```
#### quantile.factory( a, b, c )
Returns a function for evaluating the [quantile function][quantile-function] of a [triangular][triangular-distribution] distribution with parameters `a` (lower limit), `b` (upper limit) and `c` (mode).
```javascript
var myquantile = quantile.factory( 2.0, 4.0, 2.5 );
var y = myquantile( 0.4 );
// returns ~2.658
y = myquantile( 0.8 );
// returns ~3.225
```
</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/triangular/quantile' );
var a;
var b;
var c;
var p;
var y;
var i;
for ( i = 0; i < 25; i++ ) {
p = randu();
a = randu() * 10.0;
b = a + (randu() * 40.0);
c = a + ((b-a) * randu());
y = quantile( p, a, b, c );
console.log( 'p: %d, a: %d, b: %d, c: %d, Q(p;a,b,c): %d', p.toFixed( 4 ), a.toFixed( 4 ), b.toFixed( 4 ), c.toFixed( 4 ), y.toFixed( 4 ) );
}
```
</section>
<!-- /.examples -->
<section class="links">
[triangular-distribution]: https://en.wikipedia.org/wiki/Triangular_distribution
[quantile-function]: https://en.wikipedia.org/wiki/Quantile_function
</section>
<!-- /.links -->
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