Moment-Generating Function

Negative binomial distribution moment-generating function (MGF).

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

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

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 uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var mgf = require( '@stdlib/stats-base-dists-negative-binomial-mgf' );

var opts = {
    'dtype': 'float64'
};
var t = uniform( 10, -0.5, 0.5, opts );
var r = uniform( 10, 0.0, 50.0, opts );
var p = uniform( 10, 0.0, 1.0, opts );

logEachMap( 't: %0.4f, r: %0.4f, p: %0.4f, M_X(t;r,p): %0.4f', t, r, p, mgf );

C APIs

Usage

#include "stdlib/stats/base/dists/negative-binomial/mgf.h"

stdlib_base_dists_negative_binomial_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.

double out = stdlib_base_dists_negative_binomial_mgf( 0.05, 20.0, 0.8 );
// returns ~267.839

The function accepts the following arguments:

• t: [in] double input value.
• r: [in] double number of successes until experiment is stopped.
• p: [in] double success probability.

double stdlib_base_dists_negative_binomial_mgf( const double t, const double r, const double p );

Examples

#include "stdlib/stats/base/dists/negative-binomial/mgf.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 t;
    double r;
    double p;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        t = random_uniform( -1.0, 1.0 );
        r = random_uniform( 1.0, 10.0 );
        p = random_uniform( 0.0, 1.0 );
        y = stdlib_base_dists_negative_binomial_mgf( t, r, p );
        printf( "t: %lf, r: %lf, p: %lf, M_X(t;r,p): %lf\n", t, r, p, y );
    }
}

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

License

See LICENSE.

Copyright

Copyright © 2016-2026. The Stdlib Authors.