npm package: @stdlib/stats-base-dists-discrete-uniform-mgf
Description: Discrete uniform 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] -->

> [Discrete uniform][discrete-uniform-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 [discrete uniform][discrete-uniform-distribution] random variable is

<!-- <equation class="equation" label="eq:discrete_uniform_mgf" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right]= \begin{cases} \frac{\mathrm{e}^{at}-\mathrm{e}^{t(b+1)}}{(b-a+1)(1-e^t)} & \text{for } t \neq 0 \\ 1 & \text{for } t = 0 \end{cases}" alt="Moment-generating function (MGF) for a discrete uniform distribution."> -->

<div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right]= \begin{cases} \frac{\mathrm{e}^{at}-\mathrm{e}^{t(b+1)}}{(b-a+1)(1-e^t)} &amp; \text{for } t \neq 0 \\ 1 &amp; \text{for } t = 0 \end{cases}" data-equation="eq:discrete_uniform_mgf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/discrete-uniform/mgf/docs/img/equation_discrete_uniform_mgf.svg" alt="Moment-generating function (MGF) for a discrete uniform distribution.">
    <br>
</div>

<!-- </equation> -->

where `a` is the minimum support and `b` is the maximum support. The parameters must satisfy `a <= b`.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### mgf( t, a, b )

Evaluates the [moment-generating function][mgf] (MGF) for a [discrete uniform][discrete-uniform-distribution] distribution with parameters `a` (minimum support) and `b` (maximum support).

```javascript
var y = mgf( 2.0, 0, 4 );
// returns ~689.475

y = mgf( -0.2, 0, 4 );
// returns ~0.697

y = mgf( 2.0, 0, 1 );
// returns ~4.195
```

If `a` or `b` is not an integer value, the function returns `NaN`.

```javascript
var y = mgf( 0.2, 1, 5.5 );
// returns NaN
```

If provided `a > b`, the function returns `NaN`.

```javascript
var y = mgf( 0.5, 3, 2);
// returns NaN
```

If provided `NaN` for any parameter, the function returns `NaN`.

```javascript
var y = mgf( NaN, 0, 1 );
// returns NaN

y = mgf( 0.0, NaN, 1 );
// returns NaN

y = mgf( 0.0, 0, NaN );
// returns NaN
```

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

Returns a function for evaluating the [moment-generating function][mgf] (MGF) of a [discrete uniform][discrete-uniform-distribution] distribution with parameters `a` (minimum support) and `b` (maximum support).

```javascript
var mymgf = mgf.factory( 6, 7 );
var y = mymgf( 0.1 );
// returns ~1.918

y = mymgf( 1.1 );
// returns ~1471.722
```

</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 randint = require( '@stdlib/random-base-discrete-uniform' );
var randu = require( '@stdlib/random-base-randu' );
var mgf = require( '@stdlib/stats-base-dists-discrete-uniform-mgf' );

var randa = randint.factory( 0, 5 );
var randb = randint.factory();
var a;
var b;
var t;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    t = randu();
    a = randa();
    b = randb( a, a+randa() );
    v = mgf( t, a, b );
    console.log( 't: %d, a: %d, b: %d, M_X(t;a,b): %d', t.toFixed( 4 ), a.toFixed( 4 ), b.toFixed( 4 ), v.toFixed( 4 ) );
}
```

</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/discrete-uniform/mgf.h"
```

#### stdlib_base_dists_discrete_uniform_mgf( t, a, b )

Evaluates the [moment-generating function][mgf] (MGF) for a [discrete uniform][discrete-uniform-distribution] distribution with parameters `a` (minimum support) and `b` (maximum support).

```c
double out = stdlib_base_dists_discrete_uniform_mgf( 2.0, 0, 4 );
// returns ~689.475
```

The function accepts the following arguments:

-   **t**: `[in] double` input value.
-   **a**: `[in] int32_t` minimum support.
-   **b**: `[in] int32_t` maximum support.

```c
double stdlib_base_dists_discrete_uniform_mgf( const double t, const int32_t a, const int32_t 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/discrete-uniform/mgf.h"
#include "stdlib/math/base/special/round.h"
#include <stdint.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 ) {
    int32_t a;
    int32_t b;
    double t;
    double y;
    int i;

    for ( i = 0; i < 10; i++ ) {
        t = random_uniform( -10.0, 10.0 );
        a = stdlib_base_round( random_uniform( 0.0, 10.0 ) );
        b = stdlib_base_round( random_uniform( 0.0, 10.0 ) ) + a;
        y = stdlib_base_dists_discrete_uniform_mgf( t, a, b );
        printf( "t: %lf, a: %d, b: %d, M_X(t;a,b): %lf\n", t, a, b, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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<section class="references">

</section>

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<section class="related">

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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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[discrete-uniform-distribution]: https://en.wikipedia.org/wiki/Discrete_uniform_distribution

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

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