Logarithm of Probability Density Function

Rayleigh distribution logarithm of probability density function (PDF).

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

f left-parenthesis x semicolon sigma right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column StartFraction x Over sigma squared EndFraction e Superscript minus x squared slash left-parenthesis 2 sigma squared right-parenthesis Baseline 2nd Column a m p semicolon for x greater-than-or-equal-to 0 2nd Row 1st Column 0 2nd Column otherwise EndLayout

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 ) );
}