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# Probability Density Function

> [Triangular][triangular-distribution] distribution [probability density function][pdf] (PDF).

<section class="intro">

The [probability density function][pdf] (PDF) for a [triangular][triangular-distribution] random variable is

<!-- <equation class="equation" label="eq:triangular_pdf" align="center" raw="f(x;a,b,c)=\begin{cases} 0 & \text{for } x < a \\ \frac{2(x-a)}{(b-a)(c-a)} & \text{for } a \le x < c \\ \frac{2}{b-a} & \text{for } x = c \\ \frac{2(b-x)}{(b-a)(b-c)} & \text{for } c < x \le b \\ 0 & \text{for } b < x \end{cases}" alt="Probability density function (PDF) for a triangular distribution."> -->

<div class="equation" align="center" data-raw-text="f(x;a,b,c)=\begin{cases} 0 &amp; \text{for } x &lt; a \\ \frac{2(x-a)}{(b-a)(c-a)} &amp; \text{for } a \le x &lt; c \\ \frac{2}{b-a} &amp; \text{for } x = c \\ \frac{2(b-x)}{(b-a)(b-c)} &amp; \text{for } c &lt; x \le b \\ 0 &amp; \text{for } b &lt; x \end{cases}" data-equation="eq:triangular_pdf">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/stats/base/dists/triangular/pdf/docs/img/equation_triangular_pdf.svg" alt="Probability density function (PDF) for a triangular distribution.">
    <br>
</div>

<!-- </equation> -->

where `a` is the lower limit and `b` is the upper limit and `c` is the mode.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var pdf = require( '@stdlib/stats/base/dists/triangular/pdf' );
```

#### pdf( x, a, b, c )

Evaluates the [probability density function][pdf] (PDF) for a [triangular][triangular-distribution] distribution with parameters `a` (lower limit), `b` (upper limit) and `c` (mode).

```javascript
var y = pdf( 0.5, -1.0, 1.0, 0.0 );
// returns 0.5

y = pdf( 0.5, -1.0, 1.0, 0.5 );
// returns 1.0

y = pdf( -10.0, -20.0, 0.0, -2.0 );
// returns ~0.056

y = pdf( -2.0, -1.0, 1.0, 0.0 );
// returns 0.0
```

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

```javascript
var y = pdf( NaN, 0.0, 1.0, 0.5 );
// returns NaN

y = pdf( 0.0, NaN, 1.0, 0.5 );
// returns NaN

y = pdf( 0.0, 0.0, NaN, 0.5 );
// returns NaN

y = pdf( 2.0, 1.0, 0.0, NaN );
// returns NaN
```

If provided parameters not satisfying `a <= c <= b`, the function returns `NaN`.

```javascript
var y = pdf( 1.0, 1.0, 0.0, 1.5 );
// returns NaN

y = pdf( 1.0, 1.0, 0.0, -1.0 );
// returns NaN

y = pdf( 1.0, 0.0, -1.0, 0.5 );
// returns NaN
```

#### pdf.factory( a, b, c )

Returns a function for evaluating the [probability density function][pdf] (PDF) of a [triangular][triangular-distribution] distribution with parameters `a` (lower limit), `b` (upper limit) and `c` (mode).

```javascript
var mypdf = pdf.factory( 0.0, 10.0, 5.0 );
var y = mypdf( 2.0 );
// returns 0.08

y = mypdf( 12.0 );
// returns 0.0
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

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

var a;
var b;
var c;
var x;
var y;
var i;

for ( i = 0; i < 25; i++ ) {
    x = randu() * 30.0;
    a = randu() * 10.0;
    b = a + (randu() * 40.0);
    c = a + ((b-a) * randu());
    y = pdf( x, a, b, c );
    console.log( 'x: %d, a: %d, b: %d, c: %d, f(x;a,b,c): %d', x.toFixed( 4 ), a.toFixed( 4 ), b.toFixed( 4 ), c.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

<section class="links">

[pdf]: https://en.wikipedia.org/wiki/Probability_density_function

[triangular-distribution]: https://en.wikipedia.org/wiki/Triangular_distribution

</section>

<!-- /.links -->
