Moment-Generating Function

Chi-squared distribution moment-generating function (MGF).

The moment-generating function for a chi-squared 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 2 t right-parenthesis Superscript negative k slash 2 Baseline for t less-than one-half

where k is the degrees of freedom.

Usage

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

mgf( t, k )

Evaluates the moment-generating function (MGF) for a chi-squared distribution with degrees of freedom k.

var y = mgf( 0.4, 2 );
// returns ~5.0

y = mgf( -1.0, 5.0 );
// returns ~0.0642

y = mgf( 0.0, 10.0 );
// returns 1.0

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

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

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

If provided t >= 0.5, the function returns NaN.

var y = mgf( 0.8, 1.0 );
// returns NaN

If provided k < 0, the function returns NaN.

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

mgf.factory( k )

Returns a function for evaluating the moment-generating function (MGF) for a chi-squared distribution with degrees of freedom k.

var mymgf = mgf.factory( 1.0 );

var y = mymgf( 0.2 );
// returns ~1.291

y = mymgf( 0.4 );
// returns ~2.236

Examples

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

var t;
var k;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    t = randu() * 0.5;
    k = randu() * 10.0;
    y = mgf( t, k );
    console.log( 'x: %d, k: %d, M_X(t;k): %d', t.toFixed( 4 ), k.toFixed( 4 ), y.toFixed( 4 ) );
}