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Moment-Generating Function

Beta distribution moment-generating function (MGF).

The moment-generating function for a beta random variable is

$$M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = 1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r} \right) \frac{t^k}{k!}$$

where alpha > 0 is the first shape parameter and beta > 0 is the second shape parameter.

Installation

npm install @stdlib/stats-base-dists-beta-mgf

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var mgf = require( '@stdlib/stats-base-dists-beta-mgf' );

mgf( t, alpha, beta )

Evaluates the moment-generating function (MGF) for a beta distribution with parameters alpha (first shape parameter) and beta (second shape parameter).

var y = mgf( 0.5, 1.0, 1.0 );
// returns ~1.297

y = mgf( 0.5, 2.0, 4.0 );
// returns ~1.186

y = mgf( 3.0, 2.0, 2.0 );
// returns ~5.575

y = mgf( -0.8, 4.0, 4.0 );
// returns ~0.676

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

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 alpha <= 0, the function returns NaN.

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

y = mgf( 2.0, 0.0, 0.5 );
// returns NaN

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

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

y = mgf( 2.0, 0.5, 0.0 );
// returns NaN

mgf.factory( alpha, beta )

Returns a function for evaluating the moment-generating function for a beta distribution with parameters alpha (first shape parameter) and beta (second shape parameter).

var mymgf = mgf.factory( 0.5, 0.5 );

var y = mymgf( 0.8 );
// returns ~1.552

y = mymgf( 0.3 );
// returns ~1.168

Examples

var randu = require( '@stdlib/random-base-randu' );
var EPS = require( '@stdlib/constants-float64-eps' );
var mgf = require( '@stdlib/stats-base-dists-beta-mgf' );

var alpha;
var beta;
var t;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    t = randu() * 20.0;
    alpha = ( randu()*5.0 ) + EPS;
    beta = ( randu()*5.0 ) + EPS;
    v = mgf( t, alpha, beta );
    console.log( 't: %d, α: %d, β: %d, M_X(t;α,β): %d', t.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), v.toFixed( 4 ) );
}

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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.

Community

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License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.