Moment-Generating Function

Weibull distribution moment-generating function (MGF).

The moment-generating function for a Weibull 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 sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts StartFraction t Superscript n Baseline lamda Superscript n Baseline Over n factorial EndFraction normal upper Gamma left-parenthesis 1 plus StartFraction n Over k EndFraction right-parenthesis

where lambda > 0 is the scale paramater and k > 0 is the shape parameter.

Usage

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

mgf( t, k, lambda )

Evaluates the moment-generating function (MGF) for a Weibull distribution with shape parameter k and scale parameter lambda.

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

y = mgf( -1.0, 4.0, 4.0 );
// returns ~0.019

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 k <= 0, the function returns NaN.

var y = mgf( 0.2, -1.0, 0.5 );
// returns NaN

y = mgf( 0.2, 0.0, 0.5 );
// returns NaN

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

var y = mgf( 0.2, 0.5, -1.0 );
// returns NaN

y = mgf( 0.2, 0.5, 0.0 );
// returns NaN

mgf.factory( k, lambda )

Returns a function for evaluating the moment-generating function of a Weibull distribution with shape parameter k and scale parameter lambda.

var myMGF = mgf.factory( 8.0, 10.0 );

var y = myMGF( 0.8 );
// returns ~3150.149

y = myMGF( 0.08 );
// returns ~2.137s

Examples

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

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

for ( i = 0; i < 10; i++ ) {
    t = randu() * 5.0;
    lambda = ( randu() * 10.0 ) + EPS;
    k = ( randu() * 10.0 ) + EPS;
    y = mgf( t, lambda, k );
    console.log( 'x: %d, k: %d, λ: %d, M_X(t;k,λ): %d', t.toFixed( 4 ), k.toFixed( 4 ), lambda.toFixed( 4 ), y.toFixed( 4 ) );
}