Probability Density Function

Gamma distribution probability density function (PDF).

The probability density function (PDF) for a gamma random variable is

f left-parenthesis x semicolon alpha comma beta right-parenthesis equals StartFraction beta Superscript alpha Baseline Over normal upper Gamma left-parenthesis alpha right-parenthesis EndFraction x Superscript alpha minus 1 Baseline e Superscript minus beta x

where α > 0 is the shape parameter and β > 0 is the rate parameter.

Usage

var pdf = require( '@stdlib/math/base/dists/gamma/pdf' );

pdf( x, alpha, beta )

Evaluates the probability density function (PDF) for a gamma distribution with parameters alpha (shape parameter) and beta (rate parameter).

var y = pdf( 2.0, 0.5, 1.0 );
// returns ~0.054

y = pdf( 0.1, 1.0, 1.0 );
// returns ~0.904

y = pdf( -1.0, 4.0, 2.0 );
// returns 0.0

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

var y = pdf( NaN, 1.0, 1.0 );
// returns NaN

y = pdf( 0.0, NaN, 1.0 );
// returns NaN

y = pdf( 0.0, 1.0, NaN );
// returns NaN

If provided alpha < 0, the function returns NaN.

var y = pdf( 2.0, -0.5, 1.0 );
// returns NaN

If provided alpha = 0, the function evaluates the PDF of a degenerate distribution centered at 0.

var y = pdf( 2.0, 0.0, 2.0 );
// returns 0.0

y = pdf( 0.0, 0.0, 2.0 );
// returns Infinity

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

var y = pdf( 2.0, 1.0, 0.0 );
// returns NaN

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

pdf.factory( alpha, beta )

Returns a function for evaluating the PDF of a gamma distribution with parameters alpha (shape parameter) and beta (rate parameter).

var mypdf = pdf.factory( 3.0, 1.5 );

var y = mypdf( 1.0 );
// returns ~0.377

y = mypdf( 4.0 );
// returns ~0.067

Examples

var randu = require( '@stdlib/random/base/randu' );
var pdf = require( '@stdlib/math/base/dists/gamma/pdf' );

var alpha;
var beta;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 3.0;
    alpha = randu() * 5.0;
    beta = randu() * 5.0;
    y = pdf( x, alpha, beta );
    console.log( 'x: %d, α: %d, β: %d, f(x;α,β): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}