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Moment-Generating Function

Negative binomial distribution moment-generating function (MGF).

The moment-generating function for a negative binomial random variable is

$$M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \biggl(\frac{\left( 1- p \right) e^t }{1 - p e^t}\biggr)^{\!r} \text{ for }t<-\log p$$

where r > 0 is the number of failures until the experiment is stopped and 0 <= p <= 1 is the success probability.

Installation

npm install @stdlib/stats-base-dists-negative-binomial-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-negative-binomial-mgf' );

mgf( t, r, p )

Evaluates the moment-generating function for a negative binomial distribution with number of successes until experiment is stopped r and success probability p.

var y = mgf( 0.05, 20.0, 0.8 );
// returns ~267.839

y = mgf( 0.1, 20.0, 0.1 );
// returns ~9.347

While r can be interpreted as the number of successes until the experiment is stopped, the negative binomial distribution is also defined for non-integers r. In this case, r denotes shape parameter of the gamma mixing distribution.

var y = mgf( 0.1, 15.5, 0.5 );
// returns ~26.375

y = mgf( 0.5, 7.4, 0.4 );
// returns ~2675.677

If t >= -ln( p ), the function returns NaN.

var y = mgf( 0.7, 15.5, 0.5 ); // -ln( p ) = ~0.693
// returns NaN

If provided a r which is not a positive number, the function returns NaN.

var y = mgf( 0.2, 0.0, 0.5 );
// returns NaN

y = mgf( 0.2, -2.0, 0.5 );
// returns NaN

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

var y = mgf( NaN, 20.0, 0.5 );
// returns NaN

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

y = mgf( 0.0, 20.0, NaN );
// returns NaN

If provided a success probability p outside of [0,1], the function returns NaN.

var y = mgf( 0.2, 20, -1.0 );
// returns NaN

y = mgf( 0.2, 20, 1.5 );
// returns NaN

mgf.factory( r, p )

Returns a function for evaluating the moment-generating function of a negative binomial distribution with number of successes until experiment is stopped r and success probability p.

var myMGF = mgf.factory( 4.3, 0.4 );
var y = myMGF( 0.2 );
// returns ~4.696

y = myMGF( 0.4 );
// returns ~30.83

Examples

var randu = require( '@stdlib/random-base-randu' );
var round = require( '@stdlib/math-base-special-round' );
var mgf = require( '@stdlib/stats-base-dists-negative-binomial-mgf' );

var p;
var r;
var t;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    t = (randu() * 1.0) - 0.5;
    r = randu() * 50;
    p = randu();
    y = mgf( t, r, p );
    console.log( 't: %d, r: %d, p: %d, M_X(t;r,p): %d', t, r.toFixed( 4 ), p.toFixed( 4 ), y.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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Copyright

Copyright © 2016-2024. The Stdlib Authors.