npm package: @stdlib/stats-base-dists-erlang-logpdf
Description: Natural logarithm of the probability density function (PDF) for an Erlang distribution.
Author: The Stdlib Authors
Latest version: 0.2.3

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# Logarithm of Probability Density Function

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> 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.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/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="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-erlang-logpdf
```

</section>

<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 uniform = require( '@stdlib/random-array-uniform' );
var discreteUniform = require( '@stdlib/random-array-discrete-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var logpdf = require( '@stdlib/stats-base-dists-erlang-logpdf' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 20, 0.0, 10.0, opts );
var k = discreteUniform( 20, 0, 10, opts );
var lambda = uniform( 20, 0.0, 5.0, opts );

logEachMap( 'x: %0.4f, k: %d, λ: %0.4f, ln(f(x;k,λ)): %0.4f', x, k, lambda, logpdf );
```

</section>

<!-- /.examples -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

</section>

<!-- /.related -->

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<section class="main-repo" >

* * *

## Notice

This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

#### Community

[![Chat][chat-image]][chat-url]

---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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[npm-image]: http://img.shields.io/npm/v/@stdlib/stats-base-dists-erlang-logpdf.svg
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[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/stats-base-dists-erlang-logpdf/main.svg
[coverage-url]: https://codecov.io/github/stdlib-js/stats-base-dists-erlang-logpdf?branch=main

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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-base-dists-erlang-logpdf/main/LICENSE

[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>

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