Logarithm of Probability Density Function

Cauchy distribution logarithm of probability density function (logPDF).

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

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

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) );
}