Cumulative Distribution Function

Negative binomial distribution cumulative distribution function.

The cumulative distribution function for a negative binomial random variable X is

where r is the number of successes until experiment is stopped, p is the success probability in each trial and I is the lower regularized incomplete beta function. The random variable X denotes the number of failures until the r success is reached.

Installation

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

Usage

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

cdf( x, r, p )

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

var y = cdf( 5.0, 20.0, 0.8 );
// returns ~0.617

y = cdf( 21.0, 20.0, 0.5 );
// returns ~0.622

y = cdf( 5.0, 10.0, 0.4 );
// returns ~0.034

y = cdf( 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 = cdf( 21.0, 15.5, 0.5 );
// returns ~0.859

y = cdf( 5.0, 7.4, 0.4 );
// returns ~0.131

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

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

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

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

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

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

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

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

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

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

cdf.factory( r, p )

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

var mycdf = cdf.factory( 10, 0.5 );
var y = mycdf( 3.0 );
// returns ~0.046

y = mycdf( 11.0 );
// returns ~0.668

Examples

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

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

for ( i = 0; i < 10; i++ ) {
    x = randu() * 50;
    r = randu() * 50;
    p = randu();
    y = cdf( x, r, p );
    console.log( 'x: %d, r: %d, p: %d, F(x;r,p): %d', x.toFixed( 4 ), 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.

Community

License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.