Moment-Generating Function

Binomial distribution moment-generating function (MGF).

The moment-generating function for a binomial 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 left-parenthesis 1 minus p plus p e Superscript t Baseline right-parenthesis Superscript n

where the nonnegative integer n is the number of trials and 0 <= p <= 1 is the success probability.

Usage

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

mgf( t, n, p )

Evaluates the moment-generating function for a binomial distribution with number of trials n and success probability p.

var y = mgf( 0.5, 20, 0.2 );
// returns ~11.471

y = mgf( 5.0, 20, 0.2 );
// returns ~4.798e29

y = mgf( 0.9, 10, 0.4 );
// returns ~99.338

y = mgf( 0.0, 10, 0.4 );
// returns 1.0

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

var y = mgf( NaN, 20, 0.5 );
// returns NaN

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

y = mgf( 0.0, 20, NaN );
// returns NaN

If provided a number of trials n which is not a nonnegative integer, the function returns NaN.

var y = mgf( 0.2, 1.5, 0.5 );
// returns NaN

y = mgf( 0.2, -2.0, 0.5 );
// returns NaN

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

var y = mgf( 0.2, 20, -1.0 );
// returns NaN

y = mgf( 0.2, 20, 1.5 );
// returns NaN

mgf.factory( n, p )

Returns a function for evaluating the moment-generating function of a binomial distribution with number of trials n and success probability p.

var myMGF = mgf.factory( 10, 0.5 );

var y = myMGF( 0.3 );
// returns ~5.013

Examples

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

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

for ( i = 0; i < 10; i++ ) {
    t = round( randu() * 5.0 );
    n = round( randu() * 10.0 );
    p = randu();
    y = mgf( t, n, p );
    console.log( 't: %d, n: %d, p: %d, M_X(t;n,p): %d', t, n, p.toFixed( 4 ), y.toFixed( 4 ) );
}