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 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 logpdf = require( '@stdlib/math/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 ~-1.839

y = logpdf( 4.0, 3.0, 0.1 );
// returns ~-3.458

y = logpdf( 4.0, 3.0, 3.0 );
// returns ~-2.354

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 ~-1.839

y = mylogpdf( 5.0 );
// returns ~-3.817

Examples

var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/math/float64-eps' );
var logpdf = require( '@stdlib/math/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) );
}