npm package: @stdlib/stats-base-dists-negative-binomial-mgf
Description: Negative binomial distribution moment-generating function (MGF).
Author: The Stdlib Authors
Latest version: 0.2.3

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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] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> [Negative binomial][negative-binomial-distribution] distribution moment-generating function (MGF).

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

The [moment-generating function][mgf] for a [negative binomial][negative-binomial-distribution] random variable is

<!-- <equation class="equation" label="eq:negative_binomial_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{\left( 1- p \right) e^t }{1 - p e^t}\biggr)^{\!r} \text{ for }t<-\log p" alt="Moment-generating function (MGF) for a negative binomial distribution."> -->

<div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{\left( 1- p \right) e^t }{1 - p e^t}\biggr)^{\!r} \text{ for }t&lt;-\log p" data-equation="eq:negative_binomial_mgf_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/docs/img/equation_negative_binomial_mgf_function.svg" alt="Moment-generating function (MGF) for a negative binomial distribution.">
    <br>
</div>

<!-- </equation> -->

where `r > 0` is the number of failures until the experiment is stopped and `0 <= p <= 1` is the success probability.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-negative-binomial-mgf
```

</section>

<section class="usage">

## Usage

```javascript
var mgf = require( '@stdlib/stats-base-dists-negative-binomial-mgf' );
```

#### mgf( t, r, p )

Evaluates the [moment-generating function][mgf] for a [negative binomial][negative-binomial-distribution] distribution with number of successes until experiment is stopped `r` and success probability `p`.

```javascript
var y = mgf( 0.05, 20.0, 0.8 );
// returns ~267.839

y = mgf( 0.1, 20.0, 0.1 );
// returns ~9.347
```

While `r` can be interpreted as the number of successes until the experiment is stopped, the [negative binomial][negative-binomial-distribution] distribution is also defined for non-integers `r`. In this case, `r` denotes shape parameter of the [gamma mixing distribution][negative-binomial-mixture-representation].

```javascript
var y = mgf( 0.1, 15.5, 0.5 );
// returns ~26.375

y = mgf( 0.5, 7.4, 0.4 );
// returns ~2675.677
```

If `t >= -ln( p )`, the function returns `NaN`.

```javascript
var y = mgf( 0.7, 15.5, 0.5 ); // -ln( p ) = ~0.693
// returns NaN
```

If provided a `r` which is not a positive number, the function returns `NaN`.

```javascript
var y = mgf( 0.2, 0.0, 0.5 );
// returns NaN

y = mgf( 0.2, -2.0, 0.5 );
// returns NaN
```

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

```javascript
var y = mgf( NaN, 20.0, 0.5 );
// returns NaN

y = mgf( 0.0, NaN, 0.5 );
// returns NaN

y = mgf( 0.0, 20.0, NaN );
// returns NaN
```

If provided a success probability `p` outside of `[0,1]`, the function returns `NaN`.

```javascript
var y = mgf( 0.2, 20, -1.0 );
// returns NaN

y = mgf( 0.2, 20, 1.5 );
// returns NaN
```

#### mgf.factory( r, p )

Returns a function for evaluating the [moment-generating function][mgf] of a [negative binomial][negative-binomial-distribution] distribution with number of successes until experiment is stopped `r` and success probability `p`.

```javascript
var myMGF = mgf.factory( 4.3, 0.4 );
var y = myMGF( 0.2 );
// returns ~4.696

y = myMGF( 0.4 );
// returns ~30.83
```

</section>

<!-- /.usage -->

<!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- Package usage examples. -->

<section class="examples">

## Examples

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

```javascript
var uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var mgf = require( '@stdlib/stats-base-dists-negative-binomial-mgf' );

var opts = {
    'dtype': 'float64'
};
var t = uniform( 10, -0.5, 0.5, opts );
var r = uniform( 10, 0.0, 50.0, opts );
var p = uniform( 10, 0.0, 1.0, opts );

logEachMap( 't: %0.4f, r: %0.4f, p: %0.4f, M_X(t;r,p): %0.4f', t, r, p, mgf );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

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<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/negative-binomial/mgf.h"
```

#### stdlib_base_dists_negative_binomial_mgf( t, r, p )

Evaluates the [moment-generating function][mgf] for a [negative binomial][negative-binomial-distribution] distribution with number of successes until experiment is stopped `r` and success probability `p`.

```c
double out = stdlib_base_dists_negative_binomial_mgf( 0.05, 20.0, 0.8 );
// returns ~267.839
```

The function accepts the following arguments:

-   **t**: `[in] double` input value.
-   **r**: `[in] double` number of successes until experiment is stopped.
-   **p**: `[in] double` success probability.

```c
double stdlib_base_dists_negative_binomial_mgf( const double t, const double r, const double p );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/negative-binomial/mgf.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v * ( max - min ) );
}

int main( void ) {
    double t;
    double r;
    double p;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        t = random_uniform( -1.0, 1.0 );
        r = random_uniform( 1.0, 10.0 );
        p = random_uniform( 0.0, 1.0 );
        y = stdlib_base_dists_negative_binomial_mgf( t, r, p );
        printf( "t: %lf, r: %lf, p: %lf, M_X(t;r,p): %lf\n", t, r, p, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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<section class="references">

</section>

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<section class="related">

</section>

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* * *

## 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].

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---

## License

See [LICENSE][stdlib-license].


## Copyright

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

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

[negative-binomial-mixture-representation]: https://en.wikipedia.org/wiki/Negative_binomial_distribution#Gamma.E2.80.93Poisson_mixture

[negative-binomial-distribution]: https://en.wikipedia.org/wiki/Negative_binomial_distribution

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