Cumulative Distribution Function

Negative binomial distribution cumulative distribution function.

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

upper F left-parenthesis x semicolon r comma p right-parenthesis equals 1 minus upper I Subscript p Baseline left-parenthesis x plus 1 comma r right-parenthesis

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.

Usage

var cdf = require( '@stdlib/math/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/math/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 ) );
}