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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
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-mgfUsage
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 ~9.347While 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 ~2675.677If t >= -ln( p ), the function returns NaN.
var y = mgf( 0.7, 15.5, 0.5 ); // -ln( p ) = ~0.693
// returns NaNIf 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 NaNIf 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 NaNIf 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 NaNmgf.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 ~30.83Examples
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 ) );
}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.