Moment-Generating Function

Poisson distribution moment-generating function (MGF).

The moment-generating function for a Poisson random variable is

upper M Subscript upper X Baseline left-parenthesis t right-parenthesis colon equals double-struck upper E left-bracket e Superscript t upper X Baseline right-bracket equals exp left-brace mu t plus one-half sigma squared t squared right-brace

where lambda > 0 is the mean parameter.

Usage

var mgf = require( '@stdlib/math/base/dists/poisson/mgf' );

mgf( t, lambda )

Evaluates the moment-generating function (MGF) for a Poisson distribution with parameter lambda (mean).

var y = mgf( 1.0, 1.5 );
// returns ~13.163

y = mgf( 0.5, 0.5 );
// returns ~1.383

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

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

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

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

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

y = mgf( 0.1, 0.0 );
// returns NaN

mgf.factory( lambda )

Returns a function for evaluating the moment-generating function of a Poisson distribution with parameter lambda (mean).

var mymgf = mgf.factory( 2.0 );
var y = mymgf( 0.1 );
// returns ~1.234

Examples

var randu = require( '@stdlib/random/base/randu' );
var mgf = require( '@stdlib/math/base/dists/poisson/mgf' );

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

for ( i = 0; i < 10; i++ ) {
    t = randu() * 5.0;
    lambda = randu() * 5.0;
    y = mgf( t, lambda );
    console.log( 'x: %d, λ: %d, M_X(t;λ): %d', t.toFixed( 4 ), lambda.toFixed( 4 ), y.toFixed( 4 ) );
}