Moment-Generating Function

Gumbel distribution moment-generating function (MGF).

The moment-generating function for a Gumbel random variable is

upper M Subscript upper X Baseline left-parenthesis t right-parenthesis colon equals double-struck upper E left-bracket e Superscript t upper X Baseline right-bracket equals normal upper Gamma left-parenthesis 1 plus beta t right-parenthesis e Superscript mu t

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

Usage

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

mgf( t, mu, beta )

Evaluates the moment-generating function (MGF) for a Gumbel distribution with parameters mu (location parameter) and beta > 0 (scale parameter).

var y = mgf( -1.0, 0.0, 3.0 );
// returns 6.0

y = mgf( 0.1, 0.0, 3.0 );
// returns ~1.298

y = mgf( 0.0, 0.0, 1.0 );
// returns 1.0

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

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

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

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

If provided t >= 1/beta, the function returns NaN.

var y = mgf( 0.8, 0.0, 2.0 );
// returns NaN

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

var y = mgf( 0.5, 0.0, -1.0 );
// returns NaN

y = mgf( 0.5, 0.0, 0.0 );
// returns NaN

mgf.factory( mu, beta )

Returns a function for evaluating the moment-generating function of a Gumbel distribution with parameters mu (location parameter) and beta > 0 (scale parameter).

var myMGF = mgf.factory( 0.0, 3.0 );
var y = myMGF( 1.5 );
// returns ~52.343

y = myMGF( -1.0 );
// returns Infinity

Examples

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

var beta;
var mu;
var t;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    t = randu();
    mu = (randu() * 10.0) - 5.0;
    beta = randu() * 20.0;
    y = mgf( t, mu, beta );
    console.log( 't: %d, µ: %d, β: %d, M_X(t;µ,β): %d', t.toFixed( 4 ), mu.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}