Mean

Pareto (Type I) distribution expected value.

The expected value for a Pareto (Type I) random variable is

double-struck upper E left-bracket upper X right-bracket equals StartLayout Enlarged left-brace 1st Row 1st Column normal infinity 2nd Column for alpha less-than-or-equal-to 1 2nd Row 1st Column StartFraction alpha beta Over alpha minus 1 EndFraction 2nd Column for alpha greater-than 1 EndLayout

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

Usage

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

mean( alpha, beta )

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

var v = mean( 2.0, 1.0 );
// returns 2.0

v = mean( 4.0, 12.0 );
// returns 16.0

v = mean( 8.0, 2.0 );
// returns ~2.286

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

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

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

If provided 0 < alpha <= 1, the function returns +Infinity.

var v = mean( 0.8, 1.0 );
// returns Infinity

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

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

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

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

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

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

Examples

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

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 = mean( alpha, beta );
    console.log( 'α: %d, β: %d, E(X;α,β): %d', alpha.toFixed( 4 ), beta.toFixed( 4 ), v.toFixed( 4 ) );
}