Moment-generating function

Raised cosine distribution moment-generating function.

The moment-generating function for a raised cosine 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 pi squared hyperbolic sine left-parenthesis s t right-parenthesis Over s t left-parenthesis pi squared plus s squared t squared right-parenthesis EndFraction e Superscript mu t

where μ is the location parameter and s > 0 is the scale parameter.

Usage

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

mgf( t, mu, s )

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

var y = mgf( 2.0, 0.0, 3.0 );
// returns ~7.234

y = mgf( 0.5, 0.0, 1.0 );
// returns ~1.016

y = mgf( -1.0, 4.0, 2.0 );
// returns ~0.024

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, 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, s )

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

var mymgf = mgf.factory( 10.0, 2.0 );

var y = mymgf( 0.1 );
// returns ~2.725

Examples

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

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

for ( i = 0; i < 10; i++ ) {
    t = randu() * 10.0;
    mu = randu() * 10.0;
    s = randu() * 10.0;
    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 ) );
}