Probability Density Function

Cauchy distribution probability density function (PDF).

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

f left-parenthesis x semicolon gamma comma x 0 right-parenthesis equals StartFraction 1 Over pi gamma left-bracket 1 plus left-parenthesis StartFraction x minus x 0 Over gamma EndFraction right-parenthesis squared right-bracket EndFraction

where x0 is the location parameter and gamma > 0 is the scale parameter.

Usage

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

pdf( x, x0, gamma )

Evaluates the probability density function (PDF) for a Cauchy distribution with parameters x0 (location parameter) and gamma > 0 (scale parameter).

var y = pdf( 2.0, 1.0, 1.0 );
// returns ~0.159

y = pdf( 4.0, 3.0, 0.1 );
// returns ~0.0315

y = pdf( 4.0, 3.0, 3.0 );
// returns ~0.095

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

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

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

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

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

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

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

pdf.factory( x0, gamma )

Returns a function for evaluating the PDF of a Cauchy distribution with location parameter x0 and scale parameter gamma.

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

var y = mypdf( 10.0 );
// returns ~0.159

y = mypdf( 5.0 );
// returns ~0.022

Examples

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

var gamma;
var x0;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    x0 = ( randu()*10.0 ) - 5.0;
    gamma = ( randu()*20.0 ) + EPS;
    y = pdf( x, gamma, x0 );
    console.log( 'x: %d, x0: %d, γ: %d, f(x;x0,γ): %d', x.toFixed(4), x0.toFixed(4), gamma.toFixed(4), y.toFixed(4) );
}