Moment-Generating Function

Laplace distribution moment-generating function (MGF).

The moment-generating function for a Laplace (double exponential) 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 StartFraction exp left-parenthesis mu t right-parenthesis Over 1 minus b squared t squared EndFraction for StartAbsoluteValue t EndAbsoluteValue less-than 1 slash b

where mu is the location parameter and b is the scale parameter. For |t| >= 1/b, the MGF is undefined.

Usage

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

mgf( t, mu, b )

Evaluates the moment-generating function (MGF) for a Laplace (double exponential) distribution with parameters mu (location) and b (scale).

var y = mgf( 0.5, 0.0, 1.0 );
// returns ~1.333

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

y = mgf( -1.0, 4.0, 0.2 );
// returns ~0.019

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 t is not inside the interval (-1/b, 1/b), the function returns NaN.

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

y = mgf( -0.5, 0.0, 4.0 );
// returns NaN

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

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

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

mgf.factory( mu, b )

Returns a function for evaluating the moment-generating function (MGF) of a Laplace (double exponential) distribution with parameters mu and b.

var mymgf = mgf.factory( 4.0, 2.0 );

var y = mymgf( 0.2 );
// returns ~2.649

y = mymgf( 0.4 );
// returns ~13.758

Examples

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

var mu;
var b;
var t;
var y;
var i;

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