Moment-Generating Function

Beta distribution moment-generating function (MGF).

The moment-generating function for a beta random variable is

$$M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = 1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r} \right) \frac{t^k}{k!}$$

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

Usage

var mgf = require( '@stdlib/stats/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 ~1.552

y = mymgf( 0.3 );
// returns ~1.168

Examples

var uniform = require( '@stdlib/random/array/uniform' );
var logEachMap = require( '@stdlib/console/log-each-map' );
var EPS = require( '@stdlib/constants/float64/eps' );
var mgf = require( '@stdlib/stats/base/dists/beta/mgf' );

var opts = {
    'dtype': 'float64'
};
var alpha = uniform( 10, EPS, 5.0, opts );
var beta = uniform( 10, EPS, 5.0, opts );
var t = uniform( 10, 0.0, 20.0, opts );

logEachMap( 't: %0.4f, α: %0.4f, β: %0.4f, M_X(t;α,β): %0.4f', t, alpha, beta, mgf );