Probability Mass Function

Negative binomial distribution probability mass function (PMF).

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

f left-parenthesis x semicolon r comma p right-parenthesis equals upper P left-parenthesis upper X equals x semicolon r comma p right-parenthesis equals StartBinomialOrMatrix k plus r minus 1 Choose x EndBinomialOrMatrix p Superscript r Baseline left-parenthesis 1 minus p right-parenthesis Superscript x Baseline for x equals 0 comma 1 comma 2 comma ellipsis

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.

Usage

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