Moment-Generating Function

Erlang distribution moment-generating function (MGF).

The moment-generating function for an Erlang random variable is

M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \left(1 \,-\, \frac{t}{\lambda}\right)^{-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.

Installation

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

Alternatively,

To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
If you are using Deno, visit the deno branch (see README for usage intructions).
For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

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 ) );
}

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.