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https://github.com/stdlib-js/stats-base-dists-erlang-cdf

Erlang distribution cumulative distribution function (CDF).
https://github.com/stdlib-js/stats-base-dists-erlang-cdf

cdf continuous cumulative-distribution dist distribution distribution-function erlang javascript node node-js nodejs probability statistics stats stdlib univariate

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Erlang distribution cumulative distribution function (CDF).

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README 

        



  

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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] 



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



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



```math

F(x; k,\lambda) = 1 - \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^n

```



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.



## Installation



```bash

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

```



Alternatively,



-   To load the package in a website via a `script` tag without installation and bundlers, use the [ES Module][es-module] available on the [`esm`][esm-url] branch (see [README][esm-readme]).

-   If you are using Deno, visit the [`deno`][deno-url] branch (see [README][deno-readme] for usage intructions).

-   For use in Observable, or in browser/node environments, use the [Universal Module Definition (UMD)][umd] build available on the [`umd`][umd-url] branch (see [README][umd-readme]).



The [branches.md][branches-url] file summarizes the available branches and displays a diagram illustrating their relationships.



To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.



## 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

```



## Examples



```javascript

var randu = require( '@stdlib/random-base-randu' );

var round = require( '@stdlib/math-base-special-round' );

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



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 = cdf( 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 ) );

}

```



* * *



## 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 © 2016-2024. The Stdlib [Authors][stdlib-authors].



[npm-image]: http://img.shields.io/npm/v/@stdlib/stats-base-dists-erlang-cdf.svg

[npm-url]: https://npmjs.org/package/@stdlib/stats-base-dists-erlang-cdf



[test-image]: https://github.com/stdlib-js/stats-base-dists-erlang-cdf/actions/workflows/test.yml/badge.svg?branch=main

[test-url]: https://github.com/stdlib-js/stats-base-dists-erlang-cdf/actions/workflows/test.yml?query=branch:main



[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/stats-base-dists-erlang-cdf/main.svg

[coverage-url]: https://codecov.io/github/stdlib-js/stats-base-dists-erlang-cdf?branch=main



[chat-image]: https://img.shields.io/gitter/room/stdlib-js/stdlib.svg

[chat-url]: https://app.gitter.im/#/room/#stdlib-js_stdlib:gitter.im



[stdlib]: https://github.com/stdlib-js/stdlib



[stdlib-authors]: https://github.com/stdlib-js/stdlib/graphs/contributors



[umd]: https://github.com/umdjs/umd

[es-module]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Modules



[deno-url]: https://github.com/stdlib-js/stats-base-dists-erlang-cdf/tree/deno

[deno-readme]: https://github.com/stdlib-js/stats-base-dists-erlang-cdf/blob/deno/README.md

[umd-url]: https://github.com/stdlib-js/stats-base-dists-erlang-cdf/tree/umd

[umd-readme]: https://github.com/stdlib-js/stats-base-dists-erlang-cdf/blob/umd/README.md

[esm-url]: https://github.com/stdlib-js/stats-base-dists-erlang-cdf/tree/esm

[esm-readme]: https://github.com/stdlib-js/stats-base-dists-erlang-cdf/blob/esm/README.md

[branches-url]: https://github.com/stdlib-js/stats-base-dists-erlang-cdf/blob/main/branches.md



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



[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