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Logarithm of Probability Density Function

Cauchy distribution logarithm of probability density function (logPDF).

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

$$f(x;\gamma,x_0)=\frac{1}{\pi\gamma\,\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}\!$$

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

Installation

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

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

logpdf( x, x0, gamma )

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

var y = logpdf( 2.0, 1.0, 1.0 );
// returns ~-1.838

y = logpdf( 4.0, 3.0, 0.1 );
// returns ~-3.457

y = logpdf( 4.0, 3.0, 3.0 );
// returns ~-2.349

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

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

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

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

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

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

logpdf.factory( x0, gamma )

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

var mylogpdf = logpdf.factory( 10.0, 2.0 );

var y = mylogpdf( 10.0 );
// returns ~-1.838

y = mylogpdf( 5.0 );
// returns ~-3.819

Notes

• In virtually all cases, using the logpdf or logcdf functions is preferable to manually computing the logarithm of the pdf or cdf, respectively, since the latter is prone to overflow and underflow.

Examples

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

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 = logpdf( x, gamma, x0 );
    console.log( 'x: %d, x0: %d, γ: %d, ln(f(x;x0,γ)): %d', x.toFixed(4), x0.toFixed(4), gamma.toFixed(4), y.toFixed(4) );
}

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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 © 2016-2024. The Stdlib Authors.