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Cumulative Distribution Function

Cauchy distribution cumulative distribution function.

The cumulative distribution function for a Cauchy random variable is

$$F(x; x_0,\gamma)=\frac{1}{\pi} \mathop{\mathrm{arctan}} \left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}$$

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

Installation

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

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var cdf = require( '@stdlib/stats-base-dists-cauchy-cdf' );

cdf( x, x0, gamma )

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

var y = cdf( 4.0, 0.0, 2.0 );
// returns ~0.852

y = cdf( 1.0, 0.0, 2.0 );
// returns ~0.648

y = cdf( 1.0, 3.0, 2.0 );
// returns 0.25

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

var y = cdf( NaN, 0.0, 2.0 );
// returns NaN

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

y = cdf( 1.0, NaN, 3.0 );
// returns NaN

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

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

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

cdf.factory( x0, gamma )

Returns a function for evaluating the cumulative distribution function of a Cauchy distribution with parameters x0 (location parameter) and gamma > 0 (scale parameter).

var mycdf = cdf.factory( 10.0, 2.0 );

var y = mycdf( 10.0 );
// returns 0.5

y = mycdf( 12.0 );
// returns 0.75

Examples

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

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

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    x0 = randu() * 10.0;
    gamma = ( randu()*10.0 ) + EPS;
    y = cdf( x, x0, gamma );
    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 ) );
}

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C APIs

Usage

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

stdlib_base_dists_cauchy_cdf( x, x0, gamma )

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

double out = stdlib_base_dists_cauchy_cdf( 4.0, 0.0, 2.0 );
// returns ~0.852

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

Examples

#include "stdlib/stats/base/dists/cauchy/cdf.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 y;
    double x;
    int i;

    for ( i = 0; i < 25; i++ ) {
        x = random_uniform( 0.0, 10.0 );
        x0 = random_uniform( 0.0, 10.0 );
        gamma = random_uniform( 0.0, 10.0 );
        y = stdlib_base_dists_cauchy_cdf( x, x0, gamma );
        printf( "x: %lf, k: %lf, γ: %lf, F(x;x0,γ): %lf\n", x, x0, gamma, y );
    }
}

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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

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License

See LICENSE.

Copyright

Copyright © 2016-2025. The Stdlib Authors.