Probability Mass Function

Hypergeometric distribution probability mass function (PMF).

Imagine a scenario with a population of size N, of which a subpopulation of size K can be considered successes. We draw n observations from the total population. Defining the random variable X as the number of successes in the n draws, X is said to follow a hypergeometric distribution. The probability mass function (PMF) for a hypergeometric random variable is given by

f left-parenthesis x semicolon upper N comma upper K comma n right-parenthesis equals upper P left-parenthesis upper X equals x semicolon upper N comma upper K comma n right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column StartFraction StartBinomialOrMatrix upper K Choose x EndBinomialOrMatrix StartBinomialOrMatrix upper N minus upper K Choose n minus x EndBinomialOrMatrix Over StartBinomialOrMatrix upper N Choose n EndBinomialOrMatrix EndFraction 2nd Column for x equals 0 comma 1 comma 2 comma ellipsis 2nd Row 1st Column 0 2nd Column otherwise EndLayout

Usage

var pmf = require( '@stdlib/math/base/dists/hypergeometric/pmf' );

pmf( x, N, K, n )

Evaluates the probability mass function (PMF) for a hypergeometric distribution with parameters N (population size), K (subpopulation size), and n (number of draws).

var y = pmf( 1.0, 8, 4, 2 );
// returns ~0.571

y = pmf( 2.0, 8, 4, 2 );
// returns ~0.214

y = pmf( 0.0, 8, 4, 2 );
// returns ~0.214

y = pmf( 1.5, 8, 4, 2 );
// returns 0.0

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

var y = pmf( NaN, 10, 5, 2 );
// returns NaN

y = pmf( 0.0, NaN, 5, 2 );
// returns NaN

y = pmf( 0.0, 10, NaN, 2 );
// returns NaN

y = pmf( 0.0, 10, 5, NaN );
// returns NaN

If provided a population size N, subpopulation size K or draws n which is not a nonnegative integer, the function returns NaN.

var y = pmf( 2.0, 10.5, 5, 2 );
// returns NaN

y = pmf( 2.0, 10, 1.5, 2 );
// returns NaN

y = pmf( 2.0, 10, 5, -2.0 );
// returns NaN

If the number of draws n exceeds population size N, the function returns NaN.

var y = pmf( 2.0, 10, 5, 12 );
// returns NaN

y = pmf( 2.0, 8, 3, 9 );
// returns NaN

pmf.factory( N, K, n )

Returns a function for evaluating the probability mass function (PMF) of a hypergeometric distribution with parameters N (population size), K (subpopulation size), and n (number of draws).

var mypmf = pmf.factory( 30, 20, 5 );
var y = mypmf( 4.0 );
// returns ~0.34

y = mypmf( 1.0 );
// returns ~0.029

Examples

var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var pmf = require( '@stdlib/math/base/dists/hypergeometric/pmf' );

var i;
var N;
var K;
var n;
var x;
var y;

for ( i = 0; i < 10; i++ ) {
    x = round( randu() * 5.0 );
    N = round( randu() * 20.0 );
    K = round( randu() * N );
    n = round( randu() * N );
    y = pmf( x, N, K, n );
    console.log( 'x: %d, N: %d, K: %d, n: %d, P(X=x;N,K,n): %d', x, N, K, n, y.toFixed( 4 ) );
}