Probability Density Function

Lévy distribution probability density function (PDF).

The probability density function (PDF) for a Lévy random variable is

f left-parenthesis x semicolon mu comma c right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column StartRoot StartFraction c Over 2 pi EndFraction EndRoot StartFraction e Superscript minus StartFraction c Over 2 left-parenthesis x minus mu right-parenthesis EndFraction Baseline Over left-parenthesis x minus mu right-parenthesis Superscript 3 slash 2 Baseline EndFraction 2nd Column for x greater-than mu 2nd Row 1st Column 0 2nd Column otherwise EndLayout

where μ is the location parameter and c > 0 is the scale parameter.

Usage

var pdf = require( '@stdlib/math/base/dists/levy/pdf' );

pdf( x, mu, c )

Evaluates the probability density function (PDF) for a Lévy distribution with parameters mu (location parameter) and c (scale parameter).

var y = pdf( 2.0, 0.0, 1.0 );
// returns ~0.105

y = pdf( -1.0, 4.0, 4.0 );
// returns ~0.043

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

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

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

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

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

var y = pdf( 2.0, 0.0, -1.0 );
// returns NaN

pdf( 2.0, 0.0, 0.0 );
// returns NaN

pdf.factory( mu, c )

Returns a function for evaluating the probability density function (PDF) of a Lévy distribution with parameters mu (location parameter) and c (scale parameter).

var mypdf = pdf.factory( 10.0, 2.0 );

var y = mypdf( 11.0 );
// returns ~0.208

y = mypdf( 20.0 );
// returns ~0.016

Examples

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

var mu;
var c;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    mu = randu() * 10.0;
    x = ( randu()*10.0 ) + mu;
    c = ( randu()*10.0 ) + EPS;
    y = pdf( x, mu, c );
    console.log( 'x: %d, µ: %d, c: %d, f(x;µ,c): %d', x, mu, c, y );
}