@stdlib/stats/base/dists/erlang/pdf/README.md

Standard library statistical functions.

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# Probability Density Function

> [Erlang][erlang-distribution] distribution probability density function (PDF).

<section class="intro">

The [probability density function][pdf] (PDF) for an [Erlang][erlang-distribution] random variable is

<!-- <equation class="equation" label="eq:erlang_pdf" align="center" raw="f(x; k,\lambda)={\lambda^k x^{k-1} e^{-\lambda x} \over (k-1)!} 1(x \ge 0)" alt="Probability density function (PDF) for an Erlang distribution."> -->

<div class="equation" align="center" data-raw-text="f(x; k,\lambda)={\lambda^k x^{k-1} e^{-\lambda x} \over (k-1)!} 1(x \ge 0)" data-equation="eq:erlang_pdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/erlang/pdf/docs/img/equation_erlang_pdf.svg" alt="Probability density function (PDF) for an Erlang distribution.">
    <br>
</div>

<!-- </equation> -->

where `k` is the shape parameter and `lambda` is the rate parameter.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### pdf( x, k, lambda )

Evaluates the [probability density function][pdf] (PDF) for an [Erlang][erlang-distribution]  distribution with parameters `k` (shape parameter) and `lambda` (rate parameter).

```javascript
var y = pdf( 0.1, 1, 1.0 );
// returns ~0.905

y = pdf( 0.5, 2, 2.5 );
// returns ~0.895

y = pdf( -1.0, 4, 2.0 );
// returns 0.0
```

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

```javascript
var y = pdf( NaN, 1, 1.0 );
// returns NaN

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

y = pdf( 0.0, 1, NaN );
// returns NaN
```

If not provided a nonnegative integer for `k`, the function returns `NaN`.

```javascript
var y = pdf( 2.0, -2, 0.5 );
// returns NaN

y = pdf( 2.0, 0.5, 0.5 );
// returns NaN
```

If provided `k = 0`, the function evaluates the [PDF][pdf] of a [degenerate distribution][degenerate-distribution] centered at `0`.

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

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

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

```javascript
var y = pdf( 2.0, 1, 0.0 );
// returns NaN

y = pdf( 2.0, 1, -1.0 );
// returns NaN
```

#### pdf.factory( k, lambda )

Returns a `function` for evaluating the [PDF][pdf] for an [Erlang][erlang-distribution] distribution with parameters `k` (shape parameter) and `lambda` (rate parameter).

```javascript
var mypdf = pdf.factory( 3, 1.5 );

var y = mypdf( 1.0 );
// returns ~0.377

y = mypdf( 4.0 );
// returns ~0.067
```

</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 pdf = require( '@stdlib/stats/base/dists/erlang/pdf' );

var lambda;
var k;
var x;
var y;
var i;

for ( i = 0; i < 20; i++ ) {
    x = randu() * 10.0;
    k = round( randu() * 10.0 );
    lambda = randu() * 5.0;
    y = pdf( x, k, lambda );
    console.log( 'x: %d, k: %d, λ: %d, f(x;k,λ): %d', x.toFixed( 4 ), k, lambda.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>

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<section class="links">

[erlang-distribution]: https://en.wikipedia.org/wiki/Erlang_distribution

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

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

</section>

<!-- /.links -->