Moment-Generating Function

Logistic distribution moment-generating function (MGF).

The moment-generating function for a logistic 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 e Superscript mu t Baseline normal upper B left-parenthesis 1 minus s t comma 1 plus s t right-parenthesis

for st ∈ (-1,1), where mu is the location parameter and s is the scale parameter. In above equation, B denotes the Beta function. For st outside the interval (-1,1), the function is not defined.

Usage

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

mgf( t, mu, s )

Evaluates the logarithm of the moment-generating function (MGF) for a logistic distribution with parameters mu (location parameter) and s (scale parameter).

var y = mgf( 0.9, 0.0, 1.0 );
// returns ~9.15

y = mgf( 0.1, 4.0, 4.0 );
// returns ~1.971

y = mgf( -0.2, 4.0, 4.0 );
// returns ~1.921

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 s < 0 or abs( s * t ) > 1, the function returns NaN.

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

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

mgf.factory( mu, s )

Returns a function for evaluating the moment-generating function (MGF) of a logistic distribution with parameters mu (location parameter) and s (scale parameter).

var mymgf = mgf.factory( 10.0, 0.5 );

var y = mymgf( 0.5 );
// returns ~164.846

y = mymgf( 2.0 );
// returns Infinity

Examples

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

var mu;
var s;
var t;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    t = randu();
    mu = (randu() * 10.0) - 5.0;
    s = randu() * 2.0;
    y = mgf( t, mu, s );
    console.log( 't: %d, µ: %d, s: %d, M_X(t;µ,s): %d', t.toFixed( 4 ), mu.toFixed( 4 ), s.toFixed( 4 ), y.toFixed( 4 ) );
}