Variance

Pareto (Type I) distribution variance.

The variance for a Pareto (Type I) random variable is

upper V a r left-parenthesis upper X right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column normal infinity 2nd Column for alpha element-of left-parenthesis 0 comma 2 right-bracket 2nd Row 1st Column StartFraction beta squared alpha Over left-parenthesis alpha minus 1 right-parenthesis squared left-parenthesis alpha minus 2 right-parenthesis EndFraction 2nd Column for alpha greater-than 2 EndLayout

where α > 0 is the shape parameter and β > 0 is the scale parameter.

Usage

var variance = require( '@stdlib/math/base/dists/pareto-type1/variance' );

variance( alpha, beta )

Returns the variance of a Pareto (Type I) distribution with parameters alpha (shape parameter) and beta (scale parameter).

var v = variance( 2.5, 1.0 );
// returns ~2.222

v = variance( 4.0, 12.0 );
// returns 32.0

v = variance( 8.0, 2.0 );
// returns ~0.109

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

var v = variance( NaN, 2.0 );
// returns NaN

v = variance( 2.0, NaN );
// returns NaN

If provided 0 < alpha <= 2, the function returns +Infinity.

var v = variance( 0.5, 2.0 );
// returns Infinity

v = variance( 1.5, 1.0 );
// returns Infinity

If provided alpha <= 0, the function returns NaN.

var v = variance( 0.0, 1.0 );
// returns NaN

v = variance( -1.0, 1.0 );
// returns NaN

If provided beta <= 0, the function returns NaN.

var v = variance( 1.0, 0.0 );
// returns NaN

v = variance( 1.0, -1.0 );
// returns NaN

Examples

var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/math/float64-eps' );
var variance = require( '@stdlib/math/base/dists/pareto-type1/variance' );

var alpha;
var beta;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    alpha = ( randu()*10.0 ) + EPS;
    beta = ( randu()*10.0 ) + EPS;
    v = variance( alpha, beta );
    console.log( 'α: %d, β: %d, Var(X;α,β): %d', alpha.toFixed( 4 ), beta.toFixed( 4 ), v.toFixed( 4 ) );
}