Quantile Function

Negative binomial distribution quantile function.

The quantile function for a negative binomial random variable X returns for any k satisfying 0 <= k <= 1 the value x for which

upper F left-parenthesis x minus 1 semicolon r comma p right-parenthesis less-than k less-than-or-equal-to upper F left-parenthesis x semicolon r comma p right-parenthesis

holds, where F is the cumulative distribution function (CDF) of a negative binomial distribution with parameters r and p, where r is the number of successes until the experiment is stopped and p is the success probability. The random variable X denotes the number of failures until the r success is reached.

Usage

var quantile = require( '@stdlib/math/base/dists/negative-binomial/quantile' );

quantile( k, r, p )

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

var y = quantile( 0.9, 20.0, 0.2 );
// returns 106

y = quantile( 0.9, 20.0, 0.8 );
// returns 8

y = quantile( 0.5, 10.0, 0.4 );
// returns 14

y = quantile( 0.0, 10.0, 0.9 );
// returns 0

If provided an input value k outside of [0,1], the function returns NaN.

var y = quantile( 1.1, 20.0, 0.5 );
// returns NaN

y = quantile( -0.1, 20.0, 0.5 );
// returns NaN

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 = quantile( 21.0, 15.5, 0.5 );
// returns 12

y = quantile( 5.0, 7.4, 0.4 );
// returns 10

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

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

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

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

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

y = quantile( 0.3, NaN, 0.5 );
// returns NaN

y = quantile( 0.3, 20.0, NaN );
// returns NaN

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

var y = quantile( 0.3, 20.0, -1.0 );
// returns NaN

y = quantile( 0.3, 20.0, 1.5 );
// returns NaN

quantile.factory( r, p )

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

var myquantile = quantile.factory( 10.0, 0.5 );
var y = myquantile( 0.1 );
// returns 5

y = myquantile( 0.9 );
// returns 16

Examples

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

var i;
var k;
var r;
var p;
var y;

for ( i = 0; i < 10; i++ ) {
    k = randu();
    r = randu() * 100;
    p = randu();
    y = quantile( k, r, p );
    console.log( 'k: %d, r: %d, p: %d, Q(k;r,p): %d', k.toFixed( 4 ), r.toFixed( 4 ), p.toFixed( 4 ), y );
}