Probability Density Function

Pareto (Type I) distribution probability density function (PDF).

The probability density function (PDF) for a Pareto (Type I) random variable is

f left-parenthesis x semicolon alpha comma beta right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column StartFraction alpha beta Superscript alpha Baseline Over x Superscript alpha plus 1 Baseline EndFraction 2nd Column for x greater-than-or-equal-to beta 2nd Row 1st Column 0 2nd Column otherwise EndLayout

where alpha > 0 is the shape parameter and beta > 0 is the scale parameter.

Usage

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

pdf( x, alpha, beta )

Evaluates the probability density function (PDF) for a Pareto (Type I) distribution with parameters alpha (shape parameter) and beta (scale parameter).

var y = pdf( 4.0, 1.0, 1.0 );
// returns ~0.044

y = pdf( 20.0, 1.0, 10.0 );
// returns 0.025

y = pdf( 7.0, 2.0, 6.0 );
// returns ~0.21

y = pdf( 7.0, 6.0, 3.0 );
// returns ~0.005

y = pdf( 1.0, 4.0, 2.0 );
// returns 0.0

y = pdf( 1.5, 4.0, 2.0 );
// returns 0.0

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

var y = pdf( NaN, 1.0, 1.0 );
// returns NaN

y = pdf( 0.0, NaN, 1.0 );
// returns NaN

y = pdf( 0.0, 1.0, NaN );
// returns NaN

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

var y = pdf( 2.0, -1.0, 0.5 );
// returns NaN

y = pdf( 2.0, 0.0, 0.5 );
// returns NaN

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

var y = pdf( 2.0, 0.5, -1.0 );
// returns NaN

y = pdf( 2.0, 0.5, 0.0 );
// returns NaN

pdf.factory( alpha, beta )

Returns a function for evaluating the probability density function (PDF) (CDF) of a Pareto (Type I) distribution with parameters alpha (shape parameter) and beta (scale parameter).

var mypdf = pdf.factory( 0.5, 0.5 );
var y = mypdf( 0.8 );
// returns ~0.494

y = mypdf( 2.0 );
// returns ~0.125

Examples

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

var alpha;
var beta;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 8.0;
    alpha = randu() * 4.0;
    beta = randu() * 4.0;
    y = pdf( x, alpha, beta );
    console.log( 'x: %d, α: %d, β: %d, f(x;α,β): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}