Logarithm of Cumulative Distribution Function

Logistic distribution logarithm of cumulative distribution function.

The cumulative distribution function for a logistic random variable is

upper F left-parenthesis x semicolon mu comma s right-parenthesis equals StartFraction 1 Over 1 plus e Superscript minus StartFraction x minus mu Over s EndFraction Baseline EndFraction

where mu is the location parameter and s > 0 is the scale parameter.

Usage

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

logcdf( x, mu, s )

Evaluates the logarithm of the cumulative distribution function (CDF) for a logistic distribution with parameters mu (location parameter) and s (scale parameter).

var y = logcdf( 2.0, 0.0, 1.0 );
// returns ~-0.023

y = logcdf( 0.0, 0.0, 1.0 );
// returns ~-0.693

y = logcdf( -1.0, 4.0, 2.0 );
// returns ~-5.116

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

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

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

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

If provided s < 0, the function returns NaN.

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

If provided s = 0, the function evaluates the logarithm of the CDF of a degenerate distribution centered at mu.

var y = logcdf( 2.0, 8.0, 0.0 );
// returns -Infinity

y = logcdf( 8.0, 8.0, 0.0 );
// returns 0.0

y = logcdf( 10.0, 8.0, 0.0 );
// returns 0.0

logcdf.factory( mu, s )

Returns a function for evaluating the logarithm of the cumulative distribution function of a logistic distribution with parameters mu (location parameter) and s (scale parameter).

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

var y = mylogcdf( 10.0 );
// returns -0.693

y = mylogcdf( 8.0 );
// returns ~-1.839

Examples

var randu = require( '@stdlib/random/base/randu' );
var logcdf = require( '@stdlib/math/base/dists/logistic/logcdf' );

var mu;
var s;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    mu = randu() * 10.0;
    s = randu() * 10.0;
    y = logcdf( x, mu, s );
    console.log( 'x: %d, µ: %d, s: %d, ln(F(x;µ,s)): %d', x, mu, s, y );
}