Moment-Generating Function
Erlang distribution moment-generating function (MGF).
The moment-generating function for an Erlang random variable is
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/stats/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 ~9.0
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/stats/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 ) );
}