npm package: @stdlib/stats-base-dists-erlang-cdf
Description: Erlang distribution cumulative distribution function (CDF).
Author: The Stdlib Authors
Latest version: 0.2.3

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# Cumulative Distribution Function

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> [Erlang][erlang-distribution] distribution [cumulative distribution function][cdf].

<section class="intro">

The [cumulative distribution function][cdf] for a [Erlang][erlang-distribution] random variable is

<!-- <equation class="equation" label="eq:erlang_cdf" align="center" raw="F(x; k,\lambda) = 1 - \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^n" alt="Cumulative distribution function for a Erlang distribution."> -->

<div class="equation" align="center" data-raw-text="F(x; k,\lambda) = 1 - \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^n" data-equation="eq:erlang_cdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/erlang/cdf/docs/img/equation_erlang_cdf.svg" alt="Cumulative distribution function for a Erlang distribution.">
    <br>
</div>

<!-- </equation> -->

where `k` is the shape parameter and `lambda` is the rate parameter. The [Erlang][erlang-distribution] distribution is a special case of the gamma distribution, as `k` is constrained to the natural numbers.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### cdf( x, k, lambda )

Evaluates the [cumulative distribution function][cdf] (CDF) for an [Erlang][erlang-distribution] distribution with parameters `k` (shape parameter) and `lambda` (rate parameter).

```javascript
var y = cdf( 2.0, 1, 1.0 );
// returns ~0.865

y = cdf( 2.0, 3, 1.0 );
// returns ~0.323

y = cdf( -1.0, 2, 2.0 );
// returns 0.0

y = cdf( -Infinity, 4, 2.0 );
// returns 0.0

y = cdf( +Infinity, 4, 2.0 );
// returns 1.0
```

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

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

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

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

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

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

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

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

```javascript
var y = cdf( 2.0, 0.0, 2.0 );
// returns 1.0

y = cdf( -2.0, 0.0, 2.0 );
// returns 0.0

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

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

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

y = cdf( 2.0, 1, -5.0 );
// returns NaN
```

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

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

```javascript
var mycdf = cdf.factory( 2, 0.5 );

var y = mycdf( 6.0 );
// returns ~0.801

y = mycdf( 2.0 );
// returns ~0.264
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

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

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, F(x;k,λ): %0.4f', x, k, lambda, cdf );
```

</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

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---

## License

See [LICENSE][stdlib-license].


## Copyright

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

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[cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function

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

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

</section>

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