Moment-Generating Function

Erlang distribution moment-generating function (MGF).

The moment-generating function for an Erlang 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 StartFraction t Over lamda EndFraction right-parenthesis Superscript negative k

for t < lambda, where the nonnegative integer k is the shape parameter and lambda > 0 is the rate parameter of the distribution. In the case that t >= lambda, the MGF is not defined.

Usage

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

mgf( t, k, lambda )

Evaluates the moment-generating function (mgf) for an Erlang distribution with parameters k (shape parameter) and lambda (rate parameter).

var y = mgf( 0.3, 1, 1.0 );
// returns ~1.429

y = mgf( 2.0, 2, 3.0 );
// returns ~8.999

y = mgf( -1.0, 2, 2.0 );
// returns ~0.444

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

var y = mgf( NaN, 1, 1.0 );
// returns NaN

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

y = mgf( 0.0, 1, NaN );
// returns NaN

If not provided a nonnegative integer for k, the function returns NaN.

var y = mgf( 0.2, -2, 0.5 );
// returns NaN

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

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

var y = mgf( 0.2, 1, 0.0 );
// returns NaN

y = mgf( 0.2, 1, -5.0 );
// returns NaN

mgf.factory( k, lambda )

Returns a function for evaluating the moment-generating function for an Erlang distribution with parameters k (shape parameter) and lambda (rate parameter).

var myMGF = mgf.factory( 2, 0.5 );

var y = myMGF( 0.2 );
// returns ~2.778

y = myMGF( -0.5 );
// returns 0.25

Examples

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

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

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