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Moment-Generating Function

Discrete uniform distribution moment-generating function (MGF).

The moment-generating function for a discrete uniform random variable is

$$M_X(t) := \mathbb{E}\!\left[e^{tX}\right]= \begin{cases} \frac{\mathrm{e}^{at}-\mathrm{e}^{t(b+1)}}{(b-a+1)(1-e^t)} & \text{for } t \neq 0 \\ 1 & \text{for } t = 0 \end{cases}$$

where a is the minimum support and b is the maximum support. The parameters must satisfy a <= b.

Installation

npm install @stdlib/stats-base-dists-discrete-uniform-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-discrete-uniform-mgf' );

mgf( t, a, b )

Evaluates the moment-generating function (MGF) for a discrete uniform distribution with parameters a (minimum support) and b (maximum support).

var y = mgf( 2.0, 0, 4 );
// returns ~689.475

y = mgf( -0.2, 0, 4 );
// returns ~0.697

y = mgf( 2.0, 0, 1 );
// returns ~4.195

If a or b is not an integer value, the function returns NaN.

var y = mgf( 0.2, 1, 5.5 );
// returns NaN

If provided a > b, the function returns NaN.

var y = mgf( 0.5, 3, 2);
// returns NaN

If provided NaN for any parameter, the function returns NaN.

var y = mgf( NaN, 0, 1 );
// returns NaN

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

y = mgf( 0.0, 0, NaN );
// returns NaN

mgf.factory( a, b )

Returns a function for evaluating the moment-generating function (MGF) of a discrete uniform distribution with parameters a (minimum support) and b (maximum support).

var mymgf = mgf.factory( 6, 7 );
var y = mymgf( 0.1 );
// returns ~1.918

y = mymgf( 1.1 );
// returns ~1471.722

Examples

var randint = require( '@stdlib/random-base-discrete-uniform' );
var randu = require( '@stdlib/random-base-randu' );
var mgf = require( '@stdlib/stats-base-dists-discrete-uniform-mgf' );

var randa = randint.factory( 0, 5 );
var randb = randint.factory();
var a;
var b;
var t;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    t = randu();
    a = randa();
    b = randb( a, a+randa() );
    v = mgf( t, a, b );
    console.log( 't: %d, a: %d, b: %d, M_X(t;a,b): %d', t.toFixed( 4 ), a.toFixed( 4 ), b.toFixed( 4 ), v.toFixed( 4 ) );
}

C APIs

Usage

#include "stdlib/stats/base/dists/discrete-uniform/mgf.h"

stdlib_base_dists_discrete_uniform_mgf( t, a, b )

Evaluates the moment-generating function (MGF) for a discrete uniform distribution with parameters a (minimum support) and b (maximum support).

double out = stdlib_base_dists_discrete_uniform_mgf( 2.0, 0, 4 );
// returns ~689.475

The function accepts the following arguments:

• t: [in] double input value.
• a: [in] int32_t minimum support.
• b: [in] int32_t maximum support.

double stdlib_base_dists_discrete_uniform_mgf( const double t, const int32_t a, const int32_t b );

Examples

#include "stdlib/stats/base/dists/discrete-uniform/mgf.h"
#include "stdlib/math/base/special/round.h"
#include <stdint.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 ) {
    int32_t a;
    int32_t b;
    double t;
    double y;
    int i;

    for ( i = 0; i < 10; i++ ) {
        t = random_uniform( -10.0, 10.0 );
        a = stdlib_base_round( random_uniform( 0.0, 10.0 ) );
        b = stdlib_base_round( random_uniform( 0.0, 10.0 ) ) + a;
        y = stdlib_base_dists_discrete_uniform_mgf( t, a, b );
        printf( "t: %lf, a: %d, b: %d, M_X(t;a,b): %lf\n", t, a, b, y );
    }
}

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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

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Copyright

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