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# Logarithm of Probability Density Function

> Evaluate the natural logarithm of the probability density function (PDF) for an [Erlang][erlang-distribution] distribution.

<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.rawgit.com/stdlib-js/stdlib/7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/stats/base/dists/erlang/logpdf/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 logpdf = require( '@stdlib/stats/base/dists/erlang/logpdf' );
```

#### logpdf( x, k, lambda )

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

```javascript
var y = logpdf( 0.1, 1, 1.0 );
// returns ~-0.1

y = logpdf( 0.5, 2, 2.5 );
// returns ~-0.111

y = logpdf( -1.0, 4, 2.0 );
// returns -Infinity
```

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

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

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

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

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

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

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

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

```javascript
var y = logpdf( 2.0, 0.0, 2.0 );
// returns -Infinity

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

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

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

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

#### logpdf.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 mylogpdf = logpdf.factory( 3, 1.5 );

var y = mylogpdf( 1.0 );
// returns ~-0.977

y = mylogpdf( 4.0 );
// returns ~-2.704
```

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

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 = logpdf( x, k, lambda );
    console.log( 'x: %d, k: %d, λ: %d, ln(f(x;k,λ)): %d', x.toFixed( 4 ), k, lambda.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

<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 -->
