# Logarithm of Probability Density Function

Rayleigh distribution logarithm of probability density function (PDF).

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

where sigma > 0 is the scale parameter.

## Usage

var logpdf = require( '@stdlib/stats/base/dists/rayleigh/logpdf' );


#### logpdf( x, sigma )

Evaluates the logarithm of the probability density function for a Rayleigh distribution with scale parameter sigma.

var y = logpdf( 0.3, 1.0 );
// returns ~-1.249

y = logpdf( 2.0, 0.8 );
// returns ~-1.986

y = logpdf( -1.0, 0.5 );
// returns -Infinity


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

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

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


If provided sigma < 0, the function returns NaN.

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


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

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

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

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


#### logpdf.factory( sigma )

Returns a function for evaluating the logarithm of the probability density function (PDF) of a Rayleigh distribution with parameter sigma (scale parameter).

var mylogpdf = logpdf.factory( 4.0 );

var y = mylogpdf( 6.0 );
// returns ~-2.106

y = mylogpdf( 4.0 );
// returns ~-1.886


## Notes

• In virtually all cases, using the logpdf or logcdf functions is preferable to manually computing the logarithm of the pdf or cdf, respectively, since the latter is prone to overflow and underflow.

## Examples

var randu = require( '@stdlib/random/base/randu' );
var logpdf = require( '@stdlib/stats/base/dists/rayleigh/logpdf' );

var sigma;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
x = randu() * 10.0;
sigma = randu() * 10.0;
y = logpdf( x, sigma );
console.log( 'x: %d, σ: %d, f(x;σ): %d', x.toFixed( 4 ), sigma.toFixed( 4 ), y.toFixed( 4 ) );
}