Moment-Generating Function

Beta distribution moment-generating function (MGF).

The moment-generating function for a beta 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 1 plus sigma-summation Underscript k equals 1 Overscript normal infinity Endscripts left-parenthesis product Underscript r equals 0 Overscript k minus 1 Endscripts StartFraction alpha plus r Over alpha plus beta plus r EndFraction right-parenthesis StartFraction t Superscript k Baseline Over k factorial EndFraction

where alpha > 0 is the first shape parameter and beta > 0 is the second shape parameter.

Usage

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

mgf( t, alpha, beta )

Evaluates the moment-generating function (MGF) for a beta distribution with parameters alpha (first shape parameter) and beta (second shape parameter).

var y = mgf( 0.5, 1.0, 1.0 );
// returns ~1.297

y = mgf( 0.5, 2.0, 4.0 );
// returns ~1.186

y = mgf( 3.0, 2.0, 2.0 );
// returns ~5.575

y = mgf( -0.8, 4.0, 4.0 );
// returns ~0.676

If provided NaN as any argument, the function returns NaN.

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

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

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

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

var y = mgf( 2.0, -1.0, 0.5 );
// returns NaN

y = mgf( 2.0, 0.0, 0.5 );
// returns NaN

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

var y = mgf( 2.0, 0.5, -1.0 );
// returns NaN

y = mgf( 2.0, 0.5, 0.0 );
// returns NaN

mgf.factory( alpha, beta )

Returns a function for evaluating the moment-generating function for a beta distribution with parameters alpha (first shape parameter) and beta (second shape parameter).

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

var y = mymgf( 0.8 );
// returns ~0.705

y = mymgf( 0.3 );
// returns ~0.369

Examples

var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/math/float64-eps' );
var mgf = require( '@stdlib/math/base/dists/beta/mgf' );

var alpha;
var beta;
var t;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    t = randu() * 20.0;
    alpha = ( randu()*5.0 ) + EPS;
    beta = ( randu()*5.0 ) + EPS;
    v = mgf( t, alpha, beta );
    console.log( 't: %d, α: %d, β: %d, M_X(t;α,β): %d', t.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), v.toFixed( 4 ) );
}