Moment-Generating Function

Geometric distribution moment-generating function (MGF).

The moment-generating function for a geometric 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 p e Superscript t Baseline Over 1 minus left-parenthesis 1 minus p right-parenthesis e Superscript t Baseline EndFraction for t less-than minus ln left-parenthesis 1 minus p right-parenthesis

where 0 < p <= 1 is the success probability. For t >= -ln(1-p), the MGF is not defined.

Usage

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

mgf( t, p )

Evaluates the moment-generating function (MGF) of a geometric distribution with success probability p.

var y = mgf( 0.2, 0.5 );
// returns ~1.569

y = mgf( 0.4, 0.5 );
// returns ~2.936

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

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

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

If provided a success probability p outside of the interval [0,1], the function returns NaN.

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

y = mgf( 0.2, 2.0 );
// returns NaN

If t >= -ln(1-p), the function returns NaN.

var y = mgf( 0.8, 0.5 );
// returns NaN

mgf.factory( p )

Returns a function for evaluating the moment-generating function of a geometric distribution with parameter p (success probability).

var mymgf = mgf.factory( 0.8 );
var y = mymgf( -0.2 );
// returns ~0.783

Examples

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

var p;
var t;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    t = randu();
    p = randu();
    y = mgf( t, p );
    console.log( 't: %d, p: %d, M_X(t;p): %d', t, p.toFixed( 4 ), y.toFixed( 4 ) );
}