npm package: @stdlib/stats-base-dists-gamma-mgf
Description: Gamma distribution moment-generating function (MGF).
Author: The Stdlib Authors
Latest version: 0.3.1

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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] -->

> [Gamma][gamma-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 [gamma][gamma-distribution] random variable is

<!-- <equation class="equation" label="eq:gamma_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \left( 1 - \frac{t}{\beta} \right)^{-\alpha}" alt="Moment-generating function (MGF) for a gamma distribution."> -->

<div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \left( 1 - \frac{t}{\beta} \right)^{-\alpha}" data-equation="eq:gamma_mgf_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/gamma/mgf/docs/img/equation_gamma_mgf_function.svg" alt="Moment-generating function (MGF) for a gamma distribution.">
    <br>
</div>

<!-- </equation> -->

where `alpha` is the shape parameter and `beta` is the rate parameter. For `t >= beta`, the MGF is not defined.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### mgf( t, alpha, beta )

Evaluates the [moment-generating function][mgf] (MGF) for a [gamma][gamma-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (rate parameter).

```javascript
var y = mgf( 0.5, 0.5, 1.0 );
// returns ~1.414

y = mgf( 0.1, 1.0, 1.0 );
// returns ~1.111

y = mgf( -1.0, 4.0, 2.0 );
// returns ~0.198
```

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

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

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

y = mgf( 0.0, 1.0, NaN );
// returns NaN
```

If provided `t >= beta`, the function returns `NaN`.

```javascript
var y = mgf( 2.0, 4.0, 1.0 );
// returns NaN
```

If provided `alpha < 0`, the function returns `NaN`.

```javascript
var y = mgf( 2.0, -0.5, 1.0 );
// returns NaN
```

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

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

y = mgf( 2.0, 1.0, -1.0 );
// returns NaN
```

#### mgf.factory( alpha, beta )

Returns a function for evaluating the [MGF][mgf] of a [gamma][gamma-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (rate parameter).

```javascript
var mymgf = mgf.factory( 3.0, 1.5 );

var y = mymgf( 1.0 );
// returns ~27.0

y = mymgf( 0.5 );
// returns ~3.375
```

</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-gamma-mgf' );

var opts = {
    'dtype': 'float64'
};
var t = uniform( 10, 0.0, 3.0, opts );
var alpha = uniform( 10, 0.0, 5.0, opts );
var beta = uniform( 10, 0.0, 5.0, opts );

logEachMap( 't: %0.4f, α: %0.4f, β: %0.4f, M_X(t;α,β): %0.4f', t, alpha, beta, mgf );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- 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">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/gamma/mgf.h"
```

#### stdlib_base_dists_gamma_mgf( t, alpha, beta )

Evaluates the [moment-generating function][mgf] (MGF) for a [gamma][gamma-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (rate parameter).

```c
double y = stdlib_base_dists_gamma_mgf( 0.5, 0.5, 1.0 );
// returns ~1.414
```

The function accepts the following arguments:

-   **t**: `[in] double` input value.
-   **alpha**: `[in] double` shape parameter.
-   **beta**: `[in] double` rate parameter.

```c
double stdlib_base_dists_gamma_mgf( const double t, const double alpha, const double beta );
```

</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/gamma/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 alpha;
    double beta;
    double t;
    double y;
    int i;

    for ( i = 0; i < 10; i++ ) {
        t = random_uniform( 0.0, 3.0 );
        alpha = random_uniform( 0.0, 5.0 );
        beta = random_uniform( 0.0, 5.0 );
        y = stdlib_base_dists_gamma_mgf( t, alpha, beta );
        printf( "t: %lf, α: %lf, β: %lf, M_X(t;α,β): %lf\n", t, alpha, beta, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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</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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[gamma-distribution]: https://en.wikipedia.org/wiki/Gamma_distribution

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

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