Moment-Generating Function

Normal distribution moment-generating function (MGF).

The moment-generating function for a normal 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 exp left-brace mu t plus one-half sigma squared t squared right-brace

where mu is the mean and sigma > 0 is the standard deviation.

Usage

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

mgf( t, mu, sigma )

Evaluates the moment-generating function (MGF) for a normal distribution with parameters mu (mean) and sigma (standard deviation).

var y = mgf( 2.0, 0.0, 1.0 );
// returns ~7.389

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

y = mgf( -1.0, 4.0, 2.0 );
// returns ~0.1353

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 sigma <= 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, sigma )

Returns a function for evaluating the moment-generating function (MGF) of a normal distribution with parameters mu and sigma.

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

var y = mymgf( 1.0 );
// returns ~403.429

y = mymgf( 0.5 );
// returns ~12.182

Examples

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

var sigma;
var mu;
var t;
var y;
var i;

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