Probability Density Function

Rayleigh distribution 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 pdf = require( '@stdlib/math/base/dists/rayleigh/pdf' );

pdf( x, sigma )

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

var y = pdf( 0.3, 1.0 );
// returns ~0.287

y = pdf( 2.0, 0.8 );
// returns ~0.137

y = pdf( -1.0, 0.5 );
// returns 0.0

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

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

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

If provided sigma < 0, the function returns NaN.

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

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

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

y = pdf( 0.0, 0.0 );
// returns Infinity

y = pdf( 2.0, 0.0 );
// returns 0.0

pdf.factory( sigma )

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

var myPDF = pdf.factory( 4.0 );

var y = myPDF( 6.0 );
// returns ~0.122

y = myPDF( 4.0 );
// returns ~0.152

Examples

var randu = require( '@stdlib/random/base/randu' );
var pdf = require( '@stdlib/math/base/dists/rayleigh/pdf' );

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

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