Probability Mass Function

Negative binomial distribution probability mass function (PMF).

The probability mass function (PMF) for a negative binomial random variable X is

where r > 0 is the number of successes until experiment is stopped and 0 < p <= 1 is the success probability. The random variable X denotes the number of failures until the r success is reached.

Installation

npm install @stdlib/stats-base-dists-negative-binomial-pmf

Usage

var pmf = require( '@stdlib/stats-base-dists-negative-binomial-pmf' );

pmf( x, r, p )

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

var y = pmf( 5.0, 20.0, 0.8 );
// returns ~0.157

y = pmf( 21.0, 20.0, 0.5 );
// returns ~0.06

y = pmf( 5.0, 10.0, 0.4 );
// returns ~0.016

y = pmf( 0.0, 10.0, 0.9 );
// returns ~0.349

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 = pmf( 21.0, 15.5, 0.5 );
// returns ~0.037

y = pmf( 5.0, 7.4, 0.4 );
// returns ~0.051

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

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

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

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

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

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

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

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

var y = pmf( 2.0, 20, -1.0 );
// returns NaN

y = pmf( 2.0, 20, 1.5 );
// returns NaN

pmf.factory( r, p )

Returns a function for evaluating the probability mass function (PMF) of a negative binomial distribution with number of successes until experiment is stopped r and success probability p.

var mypmf = pmf.factory( 10, 0.5 );
var y = mypmf( 3.0 );
// returns ~0.03

y = mypmf( 10.0 );
// returns ~0.088

Examples

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

var i;
var r;
var p;
var x;
var y;

for ( i = 0; i < 10; i++ ) {
    x = round( randu() * 30 );
    r = randu() * 50;
    p = randu();
    y = pmf( x, r, p );
    console.log( 'x: %d, r: %d, p: %d, P(X=x;r,p): %d', x, r, 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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Copyright

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