Kurtosis

Negative binomial distribution excess kurtosis.

The excess kurtosis for a negative binomial random variable is

upper K u r t left-parenthesis upper X right-parenthesis equals StartFraction 6 Over r EndFraction plus StartFraction left-parenthesis 1 minus p right-parenthesis squared Over p r EndFraction

where r is the number of successes until experiment is stopped and p is the success probability in each trial. The random variable X denotes the number of failures until the r success is reached.

Usage

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

kurtosis( r, p )

Returns the excess kurtosis of a negative binomial distribution with parameters r (number of successes until experiment is stopped) and p (success probability).

var v = kurtosis( 100, 0.2 );
// returns ~0.061

v = kurtosis( 50, 0.5 );
// returns ~0.13

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

var v = kurtosis( NaN, 0.5 );
// returns NaN

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

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

var v = kurtosis( -2.0, 0.5 );
// returns NaN

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

var v = kurtosis( 20, -1.0 );
// returns NaN

v = kurtosis( 20, 1.5 );
// returns NaN

Examples

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

var v;
var i;
var r;
var p;

for ( i = 0; i < 10; i++ ) {
    r = randu() * 100;
    p = randu();
    v = kurtosis( r, p );
    console.log( 'r: %d, p: %d, Kurt(X;r,p): %d', r, p.toFixed( 4 ), v.toFixed( 4 ) );
}