Moment-Generating Function

Discrete uniform distribution moment-generating function (MGF).

The moment-generating function for a discrete uniform 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 StartLayout Enlarged left-brace 1st Row 1st Column StartFraction normal e Superscript a t Baseline minus normal e Superscript t left-parenthesis b plus 1 right-parenthesis Baseline Over left-parenthesis b minus a plus 1 right-parenthesis left-parenthesis 1 minus e Superscript t Baseline right-parenthesis EndFraction 2nd Column for t not-equals 0 2nd Row 1st Column 1 2nd Column for t equals 0 EndLayout

where a is the minimum support and b is the maximum support. The parameters must satisfy a <= b.

Usage

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

mgf( t, a, b )

Evaluates the moment-generating function (MGF) for a discrete uniform distribution with parameters a (minimum support) and b (maximum support).

var y = mgf( 2.0, 0, 4 );
// returns ~689.475

y = mgf( -0.2, 0, 4 );
// returns ~0.697

y = mgf( 2.0, 0, 1 );
// returns ~4.195

If a or b is not an integer value, the function returns NaN.

var y = mgf( 0.2, 1, 5.5 );
// returns NaN

If provided a > b, the function returns NaN.

var y = mgf( 0.5, 3, 2);
// returns NaN

If provided NaN for any parameter, the function returns NaN.

var y = mgf( NaN, 0, 1 );
// returns NaN

y = mgf( 0.0, NaN, 1 );
// returns NaN

y = mgf( 0.0, 0, NaN );
// returns NaN

mgf.factory( a, b )

Returns a function for evaluating the moment-generating function (MGF) of a discrete uniform distribution with parameters a (minimum support) and b (maximum support).

var mymgf = mgf.factory( 6, 7 );
var y = mymgf( 0.1 );
// returns ~1.916

y = mymgf( 1.1 );
// returns ~1339.321

Examples

var randint = require( '@stdlib/random/base/discrete-uniform' );
var randu = require( '@stdlib/random/base/randu' );
var mgf = require( '@stdlib/math/base/dists/discrete-uniform/mgf' );

var randa = randint.factory( 0, 5 );
var randb = randint.factory();
var a;
var b;
var t;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    t = randu();
    a = randa();
    b = randb( a, a+randa() );
    v = mgf( t, a, b );
    console.log( 't: %d, a: %d, b: %d, M_X(t;a,b): %d', t.toFixed( 4 ), a.toFixed( 4 ), b.toFixed( 4 ), v.toFixed( 4 ) );
}