Probability Density Function

Gumbel distribution probability density function (PDF).

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

f left-parenthesis x semicolon mu comma beta right-parenthesis equals StartFraction 1 Over beta EndFraction e Superscript minus left-parenthesis StartFraction x minus mu Over beta EndFraction plus e Super Superscript minus StartFraction x minus mu Over beta EndFraction Superscript right-parenthesis

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

Usage

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

pdf( x, mu, beta )

Evaluates the probability density function (PDF) for a Gumbel distribution with parameters mu (location parameter) and beta (scale parameter).

var y = pdf( 0.0, 0.0, 2.0 );
// returns ~0.184

y = pdf( 0.0, 0.0, 1.0 );
// returns ~0.368

y = pdf( 1.0, 3.0, 2.0 );
// returns ~0.09

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

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

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

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

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

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

y = pdf( 2.0, 8.0, 0.0 );
// returns NaN

pdf.factory( mu, beta )

Partially apply mu and beta to create a reusable function for evaluating the PDF.

var mypdf = pdf.factory( 10.0, 2.0 );

y = mypdf( 10.0 );
// returns ~0.184

y = mypdf( 12.0 );
// returns ~0.127

Examples

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

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

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    mu = randu() * 10.0;
    beta = randu() * 10.0;
    y = pdf( x, mu, beta );
    console.log( 'x: %d, µ: %d, β: %d, f(x;µ,β): %d', x, mu, beta, y );
}