Moment-Generating Function

Exponential distribution moment-generating function (MGF).

The moment-generating function for an exponential 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 lamda Over lamda minus t EndFraction comma for t less-than lamda

where lambda > 0 is the rate parameter. For t >= lambda, the MGF is undefined.

Usage

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

mgf( t, lambda )

Evaluates the moment-generating function (MGF) for an exponential distribution.

var y = mgf( 2.0, 3.0 );
// returns 3.0

y = mgf( 0.4, 1.2 );
// returns 1.5

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 lambda < 0 or t >= lambda, the function returns NaN.

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

y = mgf( 3.0, 2.0 );
// returns NaN

mgf.factory( lambda )

Returns a function for evaluating the moment-generating function of an exponential distribution with parameter lambda(rate parameter).

var mymgf = mgf.factory( 4.0 );
var y = mymgf( 3.0 );
// returns 4.0

Examples

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

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

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