Logarithm of Probability Density Function

Evaluate the natural logarithm of the probability density function (PDF) for an Erlang distribution.

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

logpdf( x, k, lambda )

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

var y = logpdf( 0.1, 1, 1.0 );
// returns ~-0.1

y = logpdf( 0.5, 2, 2.5 );
// returns ~-0.111

y = logpdf( -1.0, 4, 2.0 );
// returns -Infinity

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

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

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

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

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

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

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

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

var y = logpdf( 2.0, 0.0, 2.0 );
// returns -Infinity

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

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

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

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

logpdf.factory( k, lambda )

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

var mylogpdf = logpdf.factory( 3, 1.5 );

var y = mylogpdf( 1.0 );
// returns ~-0.976

y = mylogpdf( 4.0 );
// returns ~-2.703

Examples

var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var logpdf = require( '@stdlib/math/base/dists/erlang/logpdf' );

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 = logpdf( x, k, lambda );
    console.log( 'x: %d, k: %d, λ: %d, ln(f(x;k,λ)): %d', x.toFixed( 4 ), k, lambda.toFixed( 4 ), y.toFixed( 4 ) );
}