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

Erlang distribution moment-generating function (MGF).
https://github.com/stdlib-js/stats-base-dists-erlang-mgf

continuous dist distribution erlang generating-functions javascript mgf moments node node-js nodejs statistics stats stdlib univariate

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Erlang distribution moment-generating function (MGF).

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README 

        



  

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# Moment-Generating Function



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



> [Erlang][erlang-distribution] distribution moment-generating function (MGF).



The [moment-generating function][mgf] for an [Erlang][erlang-distribution] random variable is



```math

M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \left(1 \,-\, \frac{t}{\lambda}\right)^{-k}

```



for `t < lambda`, where the nonnegative integer `k` is the shape parameter and `lambda > 0` is the rate parameter of the distribution. In the case that `t >= lambda`, the MGF is not defined.



## Installation



```bash

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

```



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

```



#### mgf( t, k, lambda )



Evaluates the [moment-generating function][mgf] (mgf) for an [Erlang][erlang-distribution] distribution with parameters `k` (shape parameter) and `lambda` (rate parameter).



```javascript

var y = mgf( 0.3, 1, 1.0 );

// returns ~1.429



y = mgf( 2.0, 2, 3.0 );

// returns ~9.0



y = mgf( -1.0, 2, 2.0 );

// returns ~0.444

```



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



```javascript

var y = mgf( NaN, 1, 1.0 );

// returns NaN



y = mgf( 0.0, NaN, 1.0 );

// returns NaN



y = mgf( 0.0, 1, NaN );

// returns NaN

```



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



```javascript

var y = mgf( 0.2, -2, 0.5 );

// returns NaN



y = mgf( 0.2, 0.5, 0.5 );

// returns NaN

```



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



```javascript

var y = mgf( 0.2, 1, 0.0 );

// returns NaN



y = mgf( 0.2, 1, -5.0 );

// returns NaN

```



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



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



```javascript

var myMGF = mgf.factory( 2, 0.5 );



var y = myMGF( 0.2 );

// returns ~2.778



y = myMGF( -0.5 );

// returns 0.25

```



## Examples



```javascript

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

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

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



var lambda;

var k;

var t;

var y;

var i;



for ( i = 0; i < 10; i++ ) {

    k = round( randu() * 10.0 );

    lambda = randu() * 10.0;

    t = randu() * lambda;

    y = mgf( t, k, lambda );

    console.log( 't: %d, k: %d, λ: %d, M_X(t;k,λ): %d', t.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-mgf.svg

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



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

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



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

[coverage-url]: https://codecov.io/github/stdlib-js/stats-base-dists-erlang-mgf?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-mgf/tree/deno

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

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

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

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

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

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



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



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



[mgf]: https://en.wikipedia.org/wiki/Moment-generating_function