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About stdlib...

We believe in a future in which the web is a preferred environment for numerical computation. To help realize this future, we've built stdlib. stdlib is a standard library, with an emphasis on numerical and scientific computation, written in JavaScript (and C) for execution in browsers and in Node.js.

The library is fully decomposable, being architected in such a way that you can swap out and mix and match APIs and functionality to cater to your exact preferences and use cases.

When you use stdlib, you can be absolutely certain that you are using the most thorough, rigorous, well-written, studied, documented, tested, measured, and high-quality code out there.

To join us in bringing numerical computing to the web, get started by checking us out on GitHub, and please consider financially supporting stdlib. We greatly appreciate your continued support!

Moment-Generating Function

NPM version Build Status Coverage Status

Discrete uniform distribution moment-generating function (MGF).

The moment-generating function for a discrete uniform random variable is

$$M_X(t) := \mathbb{E}\!\left[e^{tX}\right]= \begin{cases} \frac{\mathrm{e}^{at}-\mathrm{e}^{t(b+1)}}{(b-a+1)(1-e^t)} & \text{for } t \neq 0 \\ 1 & \text{for } t = 0 \end{cases}$$

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

Usage

To use in Observable,

mgf = require( 'https://cdn.jsdelivr.net/gh/stdlib-js/stats-base-dists-discrete-uniform-mgf@umd/browser.js' )

To vendor stdlib functionality and avoid installing dependency trees for Node.js, you can use the UMD server build:

var mgf = require( 'path/to/vendor/umd/stats-base-dists-discrete-uniform-mgf/index.js' )

To include the bundle in a webpage,

<script type="text/javascript" src="https://cdn.jsdelivr.net/gh/stdlib-js/stats-base-dists-discrete-uniform-mgf@umd/browser.js"></script>

If no recognized module system is present, access bundle contents via the global scope:

<script type="text/javascript">
(function () {
    window.mgf;
})();
</script>

mgf( t, a, b )

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

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.

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

If provided a > b, the function returns NaN.

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

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

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) of a discrete uniform distribution with parameters a (minimum support) and b (maximum support).

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

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

Examples

<!DOCTYPE html>
<html lang="en">
<body>
<script type="text/javascript" src="https://cdn.jsdelivr.net/gh/stdlib-js/random-base-discrete-uniform@umd/browser.js"></script>
<script type="text/javascript" src="https://cdn.jsdelivr.net/gh/stdlib-js/random-base-randu@umd/browser.js"></script>
<script type="text/javascript" src="https://cdn.jsdelivr.net/gh/stdlib-js/stats-base-dists-discrete-uniform-mgf@umd/browser.js"></script>
<script type="text/javascript">
(function () {

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 ) );
}

})();
</script>
</body>
</html>

Notice

This package is part of 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, see the main project repository.

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License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.