Moment-Generating Function

Exponential distribution moment-generating function (MGF).

The moment-generating function for an exponential random variable is

$$M_X(t):=\mathbb{E}\left[e^{tX}\right]=\frac{\lambda }{\lambda -t},\text{ for }t<\lambda$$

where lambda > 0 is the rate parameter. For t >= lambda, the MGF is undefined.

Usage

var mgf = require( '@stdlib/stats/base/dists/exponential/mgf' );

mgf( t, lambda )

Evaluates the moment-generating function (MGF) for an exponential distribution.

var y = mgf( 2.0, 3.0 );
// returns 3.0

y = mgf( 0.4, 1.2 );
// returns 1.5

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

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

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

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

var y = mgf( -2.0, -1.0 );
// returns NaN

y = mgf( 3.0, 2.0 );
// returns NaN

mgf.factory( lambda )

Returns a function for evaluating the moment-generating function of an exponential distribution with parameter lambda(rate parameter).

var mymgf = mgf.factory( 4.0 );
var y = mymgf( 3.0 );
// returns 4.0

Examples

var randu = require( '@stdlib/random/base/randu' );
var mgf = require( '@stdlib/stats/base/dists/exponential/mgf' );

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

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

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C APIs

Usage

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

stdlib_base_dists_exponential_mgf( t, lambda )

Evaluates the moment-generating function (MGF) for an exponential distribution.

double out = stdlib_base_dists_exponential_mgf( 2.0, 3.0 );
// returns 3.0

The function accepts the following arguments:

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

double stdlib_base_dists_exponential_mgf( const double t, const double lambda );

Examples

#include "stdlib/stats/base/dists/exponential/mgf.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 t;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        t = random_uniform( -1.0, 1.0 );
        lambda = random_uniform( 1.1, 10.0 );
        y = stdlib_base_dists_exponential_mgf( t, lambda );
        printf( "t: %lf, λ: %lf, M_X(t;λ): %lf\n", t, lambda, y );
    }
}