npm package: @stdlib/stats-base-dists-normal-mgf
Description: Normal 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] -->

> [Normal][normal-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 [normal][normal-distribution] random variable is

<!-- <equation class="equation" label="eq:normal_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \exp\{ \mu t + \frac{1}{2}\sigma^2t^2 \}" alt="Moment-generating function (MGF) for a normal distribution."> -->

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

<!-- </equation> -->

where `mu` is the mean and `sigma > 0` is the standard deviation.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### mgf( t, mu, sigma )

Evaluates the [moment-generating function][mgf] (MGF) for a [normal][normal-distribution] distribution with parameters `mu` (mean) and `sigma` (standard deviation).

```javascript
var y = mgf( 2.0, 0.0, 1.0 );
// returns ~7.389

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

y = mgf( -1.0, 4.0, 2.0 );
// returns ~0.1353
```

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 provided `sigma <= 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, sigma )

Returns a function for evaluating the [moment-generating function][mgf] (MGF) of a [normal][normal-distribution] distribution with parameters `mu` and `sigma`.

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

var y = mymgf( 1.0 );
// returns ~403.429

y = mymgf( 0.5 );
// returns ~12.182
```

</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-normal-mgf' );

var opts = {
    'dtype': 'float64'
};
var sigma = uniform( 10, 0.0, 20.0, opts );
var mu = uniform( 10, -5.0, 5.0, opts );
var t = uniform( 10, 0.0, 10.0, opts );

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

#### stdlib_base_dists_normal_mgf( t, mu, sigma )

Evaluates the [moment-generating function][mgf] (MGF) for a [normal][normal-distribution] distribution with parameters `mu` (mean) and `sigma` (standard deviation).

```c
double y = stdlib_base_dists_normal_mgf( 2.0, 0.0, 1.0 );
// returns ~7.389
```

The function accepts the following arguments:

-   **t**: `[in] double` input value.
-   **mu**: `[in] double` mean.
-   **sigma**: `[in] double` standard deviation.

```c
double stdlib_base_dists_normal_mgf( const double t, const double mu, const double sigma );
```

</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/normal/mgf.h"
#include "stdlib/constants/float64/eps.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 sigma;
    double mu;
    double t;
    double y;
    int i;

    for ( i = 0; i < 10; i++ ) {
        t = random_uniform( 0.0, 1.0 );
        mu = random_uniform( -50.0, 50.0 );
        sigma = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 20.0 );
        y = stdlib_base_dists_normal_mgf( t, mu, sigma );
        printf( "t: %lf, µ: %lf, σ: %lf, M_X(t;µ,σ): %lf\n", t, mu, sigma, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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</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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[normal-distribution]: https://en.wikipedia.org/wiki/Normal_distribution

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

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