npm package: @stdlib/stats-base-dists-laplace-mgf
Description: Laplace (Double Exponential) 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] -->

> [Laplace][laplace-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 [Laplace (double exponential)][laplace-distribution] random variable is

<!-- <equation class="equation" label="eq:laplace_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \frac{\exp(\mu\,t)}{1-b^2\,t^2}\,\!\text{ for }|t|<1/b" alt="Moment-generating function (MGF) for a Laplace (double exponential) distribution."> -->

<div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \frac{\exp(\mu\,t)}{1-b^2\,t^2}\,\!\text{ for }|t|&lt;1/b" data-equation="eq:laplace_mgf_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/laplace/mgf/docs/img/equation_laplace_mgf_function.svg" alt="Moment-generating function (MGF) for a Laplace (double exponential) distribution.">
    <br>
</div>

<!-- </equation> -->

where `mu` is the location parameter and `b` is the scale parameter. For `|t| >= 1/b`, the [MGF][mgf] is undefined.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### mgf( t, mu, b )

Evaluates the [moment-generating function][mgf] (MGF) for a [Laplace][laplace-distribution] (double exponential) distribution with parameters `mu` (location) and `b` (scale).

```javascript
var y = mgf( 0.5, 0.0, 1.0 );
// returns ~1.333

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

y = mgf( -1.0, 4.0, 0.2 );
// returns ~0.019
```

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

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

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

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

If `t` is not inside the interval `(-1/b, 1/b)`, the function returns `NaN`.

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

y = mgf( -0.5, 0.0, 4.0 );
// returns NaN
```

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

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

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

#### mgf.factory( mu, b )

Returns a function for evaluating the [moment-generating function][mgf] (MGF) of a [Laplace][laplace-distribution] (double exponential) distribution with parameters `mu` and `b`.

```javascript
var mymgf = mgf.factory( 4.0, 2.0 );

var y = mymgf( 0.2 );
// returns ~2.649

y = mymgf( 0.4 );
// returns ~13.758
```

</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-laplace-mgf' );

var opts = {
    'dtype': 'float64'
};
var t = uniform( 10, 0.0, 1.0, opts );
var mu = uniform( 10, -5.0, 5.0, opts );
var b = uniform( 10, 0.0, 20.0, opts );

logEachMap( 't: %0.4f, µ: %0.4f, b: %0.4f, M_X(t;µ,b): %0.4f', t, mu, b, 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/laplace/mgf.h"
```

#### stdlib_base_dists_laplace_mgf( t, mu, b )

Evaluates the [moment-generating function][mgf] (MGF) for a [Laplace][laplace-distribution] (double exponential) distribution with parameters `mu` (location) and `b` (scale).

```c
double out = stdlib_base_dists_laplace_mgf( 0.5, 0.0, 1.0 );
// returns ~1.333
```

The function accepts the following arguments:

-   **t**: `[in] double` input value.
-   **mu**: `[in] double` location parameter.
-   **b**: `[in] double` scale parameter.

```c
double stdlib_base_dists_laplace_mgf( const double t, const double mu, const double b );
```

</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/laplace/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 mu;
    double b;
    double t;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        mu = random_uniform( -5.0, 5.0 );
        b = random_uniform( 0.1, 20.0 );
        t = random_uniform( -1.0/b, 1.0/b );
        y = stdlib_base_dists_laplace_mgf( t, mu, b );
        printf( "t: %lf, µ: %lf, b: %lf, M_X(t;µ,b): %lf\n", t, mu, b, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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</section>

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* * *

## Notice

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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

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