Probability Density Function

Erlang distribution probability density function (PDF).

The probability density function (PDF) for an Erlang random variable is

f left-parenthesis x semicolon k comma lamda right-parenthesis equals StartFraction lamda Superscript k Baseline x Superscript k minus 1 Baseline e Superscript minus lamda x Baseline Over left-parenthesis k minus 1 right-parenthesis factorial EndFraction 1 left-parenthesis x greater-than-or-equal-to 0 right-parenthesis

where k is the shape parameter and lambda is the rate parameter.

Usage

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

pdf( x, k, lambda )

Evaluates the probability density function (PDF) for an Erlang distribution with parameters k (shape parameter) and lambda (rate parameter).

var y = pdf( 0.1, 1, 1.0 );
// returns ~0.904

y = pdf( 0.5, 2, 2.5 );
// returns 0.895

y = pdf( -1.0, 4, 2.0 );
// returns 0.0

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

var y = pdf( NaN, 1, 1.0 );
// returns NaN

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

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

If not provided a nonnegative integer for k, the function returns NaN.

var y = pdf( 2.0, -2, 0.5 );
// returns NaN

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

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

var y = pdf( 2.0, 1, 0.0 );
// returns NaN

y = pdf( 2.0, 1, -1.0 );
// returns NaN

pdf.factory( k, lambda )

Returns a function for evaluating the PDF for an Erlang distribution with parameters k (shape parameter) and lambda (rate parameter).

var mypdf = pdf.factory( 3, 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 round = require( '@stdlib/math/base/special/round' );
var pdf = require( '@stdlib/math/base/dists/erlang/pdf' );

var lambda;
var k;
var x;
var y;
var i;

for ( i = 0; i < 20; i++ ) {
    x = randu() * 10.0;
    k = round( randu() * 10.0 );
    lambda = randu() * 5.0;
    y = pdf( x, k, lambda );
    console.log( 'x: %d, k: %d, λ: %d, f(x;k,λ): %d', x.toFixed( 4 ), k, lambda.toFixed( 4 ), y.toFixed( 4 ) );
}