Probability Density Function

Cauchy distribution probability density function (PDF).

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

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

Installation

npm install @stdlib/stats-base-dists-cauchy-pdf

Usage

var pdf = require( '@stdlib/stats-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 uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var EPS = require( '@stdlib/constants-float64-eps' );
var pdf = require( '@stdlib/stats-base-dists-cauchy-pdf' );

var opts = {
    'dtype': 'float64'
};
var gamma = uniform( 10, EPS, 20.0, opts );
var x0 = uniform( 10, -5.0, 5.0, opts );
var x = uniform( 10, 0.0, 10.0, opts );

logEachMap( 'x: %0.4f, x0: %0.4f, γ: %0.4f, f(x;x0,γ): %0.4f', x, x0, gamma, pdf );

C APIs

Usage

#include "stdlib/stats/base/dists/cauchy/pdf.h"

stdlib_base_dists_cauchy_pdf( x, x0, gamma )

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

double out = stdlib_base_dists_cauchy_pdf( 0.5, 0.0, 2.0 );
// returns ~0.333

The function accepts the following arguments:

• x: [in] double input value.
• x0: [in] double location parameter.
• gamma: [in] double scale parameter.

double stdlib_base_dists_cauchy_pdf( const double x, const double x0, const double gamma );

Examples

#include "stdlib/stats/base/dists/cauchy/pdf.h"
#include "stdlib/constants/float64/eps.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v*(max-min) );
}

int main( void ) {
    double gamma;
    double x0;
    double x;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        x = random_uniform( 0.0, 10.0 );
        x0 = random_uniform( -5.0, 5.0 );
        gamma = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 20.0 );
        y = stdlib_base_dists_cauchy_pdf( x, x0, gamma );
        printf( "x: %lf, k: %lf, γ: %lf, f(x;x0,γ): %lf\n", x, x0, gamma, y );
    }
}

Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

License

See LICENSE.

Copyright

Copyright © 2016-2026. The Stdlib Authors.