# Logarithm of Cumulative Distribution Function

Gumbel distribution logarithm of cumulative distribution function.

The cumulative distribution function for a Gumbel random variable is

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

## Usage

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


#### logcdf( x, mu, beta )

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

var y = logcdf( 10.0, 0.0, 3.0 );
// returns ~-0.036

y = logcdf( -2.0, 0.0, 3.0 );
// returns ~-1.945

y = logcdf( 0.0, 0.0, 1.0 );
// returns ~-1


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 beta <= 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( mu, beta )

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

var mylogcdf = logcdf.factory( 0.0, 3.0 );

var y = mylogcdf( 10.0 );
// returns ~-0.036

y = mylogcdf( -2.0 );
// returns ~-1.945


## Examples

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

var beta;
var mu;
var x;
var y;
var i;

for ( i = 0; i < 100; i++ ) {
x = randu() * 10.0;
mu = randu() * 10.0;
beta = randu() * 10.0;
y = logcdf( x, mu, beta );
console.log( 'x: %d, µ: %d, β: %d, ln(F(x;µ,β)): %d', x.toFixed( 4 ), mu.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}