Logarithm of Cumulative Distribution Function

Cauchy distribution logarithm of cumulative distribution function.

The cumulative distribution function for a Cauchy random variable is

upper F left-parenthesis x semicolon x 0 comma gamma right-parenthesis equals StartFraction 1 Over pi EndFraction arc tangent left-parenthesis StartFraction x minus x 0 Over gamma EndFraction right-parenthesis plus one-half

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

Usage

var logcdf = require( '@stdlib/math/base/dists/cauchy/logcdf' );

logcdf( x, x0, gamma )

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

var y = logcdf( 4.0, 0.0, 2.0 );
// returns ~-0.16

y = logcdf( 1.0, 0.0, 2.0 );
// returns ~-0.434

y = logcdf( 1.0, 3.0, 2.0 );
// returns ~-1.386

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

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

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

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

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

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

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

logcdf.factory( x0, gamma )

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

var mylogcdf = logcdf.factory( 10.0, 2.0 );

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

y = mylogcdf( 12.0 );
// returns 0.75

Examples

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

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 = logcdf( x, x0, gamma );
    console.log( 'x: %d, x0: %d, γ: %d, ln(F(x;x0,γ)): %d', x, x0, gamma, y );
}