# 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

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 ) );
}