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.

Installation

npm install @stdlib/stats-base-dists-erlang-mgf

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 ) );
}

C APIs

Usage

#include "stdlib/stats/base/dists/erlang/mgf.h"

stdlib_base_dists_erlang_mgf( t, k, lambda )

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

double y = stdlib_base_dists_erlang_mgf( 0.3, 1, 1.0 );
// returns ~1.429

The function accepts the following arguments:

• t: [in] double input value.
• k: [in] double shape parameter.
• lambda: [in] double rate parameter.

double stdlib_base_dists_erlang_mgf( const double t, const double k, const double lambda );

Examples

#include "stdlib/stats/base/dists/erlang/mgf.h"
#include "stdlib/math/base/special/round.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v*(max-min) );
}

int main( void ) {
    double lambda;
    double k;
    double t;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        k = stdlib_base_round( random_uniform( 0.0, 10.0 ) );
        lambda = random_uniform( 0.0, 10.0 );
        t = random_uniform( 0.0, lambda );
        y = stdlib_base_dists_erlang_mgf( t, k, lambda );
        printf( "t: %lf, k: %lf, λ: %lf, M_X(t;k,λ): %lf\n", t, k, lambda, y );
    }
}

Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

License

See LICENSE.

Copyright

Copyright © 2016-2026. The Stdlib Authors.