Kurtosis

Hypergeometric distribution excess kurtosis.

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 excess kurtosis for a hypergeometric random variable is

upper K u r t left-parenthesis upper X right-parenthesis equals StartFraction left-parenthesis upper N minus 1 right-parenthesis upper N squared left-bracket upper N left-parenthesis upper N plus 1 right-parenthesis minus 6 upper K left-parenthesis upper N minus upper K right-parenthesis minus 6 n left-parenthesis upper N minus n right-parenthesis right-bracket plus 6 n upper K left-parenthesis upper N minus upper K right-parenthesis left-parenthesis upper N minus n right-parenthesis left-parenthesis 5 upper N minus 6 right-parenthesis Over n upper K left-parenthesis upper N minus upper K right-parenthesis left-parenthesis upper N minus n right-parenthesis left-parenthesis upper N minus 2 right-parenthesis left-parenthesis upper N minus 3 right-parenthesis EndFraction

Usage

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

kurtosis( N, K, n )

Returns the excess kurtosis of a hypergeometric distribution with parameters N (population size), K (subpopulation size), and n (number of draws).

var v = kurtosis( 16, 11, 4 );
// returns ~-0.326

v = kurtosis( 4, 2, 2 );
// returns 0.0

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

var v = kurtosis( NaN, 10, 4 );
// returns NaN

v = kurtosis( 20, NaN, 4 );
// returns NaN

v = kurtosis( 20, 10, 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 v = kurtosis( 10.5, 5, 2 );
// returns NaN

v = kurtosis( 10, 1.5, 2 );
// returns NaN

v = kurtosis( 10, 5, -2.0 );
// returns NaN

If the number of draws n or the subpopulation size K exceed population size N, the function returns NaN.

var v = kurtosis( 10, 5, 12 );
// returns NaN

v = kurtosis( 10, 12, 5 );
// returns NaN

Examples

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

var v;
var i;
var N;
var K;
var n;

for ( i = 0; i < 10; i++ ) {
    N = round( randu() * 20 );
    K = round( randu() * N );
    n = round( randu() * K );
    v = kurtosis( N, K, n );
    console.log( 'N: %d, K: %d, n: %d, Kurt(X;N,K,n): %d', N, K, n, v.toFixed( 4 ) );
}