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Probability Density Function

Rayleigh distribution probability density function (PDF).

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

$$f(x;\sigma) = \begin{cases} \frac{x}{\sigma^2} e^{-x^2/(2\sigma^2)} & \text{ for } x \ge 0 \\ 0 & \text{ otherwise } \end{cases}$$

where sigma > 0 is the scale parameter.

Installation

npm install @stdlib/stats-base-dists-rayleigh-pdf

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var pdf = require( '@stdlib/stats-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/stats-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 ) );
}

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C APIs

Usage

#include "stdlib/stats/base/dists/rayleigh/pdf.h"

stdlib_base_dists_rayleigh_pdf( x, sigma )

Evaluates the probability density function (PDF) for a Rayleigh distribution.

double out = stdlib_base_dists_rayleigh_pdf( 0.3, 1.0 );
// returns ~0.287

The function accepts the following arguments:

• x: [in] double input value.
• sigma: [in] double scale parameter.

double stdlib_base_dists_rayleigh_pdf( const double x, const double sigma );

Examples

#include "stdlib/stats/base/dists/rayleigh/pdf.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v*(max-min) );
}

int main( void ) {
    double sigma;
    double x;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        x = random_uniform( 0.0, 10.0 );
        sigma = random_uniform( 0.0, 10.0 );
        y = stdlib_base_dists_rayleigh_pdf( x, sigma );
        printf( "x: %lf, σ: %lf, f(x;σ): %lf\n", x, sigma, y );
    }
}

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Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

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License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.