@stdlib/stats/base/dists/rayleigh/README.md

Standard library statistical functions.

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# Rayleigh

> Rayleigh distribution.

<section class="usage">

## Usage

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

#### rayleigh

Rayleigh distribution.

```javascript
var dist = rayleigh;
// returns {...}
```

The namespace contains the following distribution functions:

<!-- <toc pattern="*+(cdf|pdf|mgf|quantile)*"> -->

<div class="namespace-toc">

-   <span class="signature">[`cdf( x, sigma )`][@stdlib/stats/base/dists/rayleigh/cdf]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution cumulative distribution function.</span>
-   <span class="signature">[`logcdf( x, sigma )`][@stdlib/stats/base/dists/rayleigh/logcdf]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution logarithm of cumulative distribution function.</span>
-   <span class="signature">[`logpdf( x, sigma )`][@stdlib/stats/base/dists/rayleigh/logpdf]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution logarithm of probability density function (PDF).</span>
-   <span class="signature">[`mgf( t, sigma )`][@stdlib/stats/base/dists/rayleigh/mgf]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution moment-generating function (MGF).</span>
-   <span class="signature">[`pdf( x, sigma )`][@stdlib/stats/base/dists/rayleigh/pdf]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution probability density function (PDF).</span>
-   <span class="signature">[`quantile( p, sigma )`][@stdlib/stats/base/dists/rayleigh/quantile]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution quantile function.</span>

</div>

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The namespace contains the following functions for calculating distribution properties:

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<div class="namespace-toc">

-   <span class="signature">[`entropy( sigma )`][@stdlib/stats/base/dists/rayleigh/entropy]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution differential entropy.</span>
-   <span class="signature">[`kurtosis( sigma )`][@stdlib/stats/base/dists/rayleigh/kurtosis]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution excess kurtosis.</span>
-   <span class="signature">[`mean( sigma )`][@stdlib/stats/base/dists/rayleigh/mean]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution expected value.</span>
-   <span class="signature">[`median( sigma )`][@stdlib/stats/base/dists/rayleigh/median]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution median.</span>
-   <span class="signature">[`mode( sigma )`][@stdlib/stats/base/dists/rayleigh/mode]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution mode.</span>
-   <span class="signature">[`skewness( sigma )`][@stdlib/stats/base/dists/rayleigh/skewness]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution skewness.</span>
-   <span class="signature">[`stdev( sigma )`][@stdlib/stats/base/dists/rayleigh/stdev]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution standard deviation.</span>
-   <span class="signature">[`variance( sigma )`][@stdlib/stats/base/dists/rayleigh/variance]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution variance.</span>

</div>

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The namespace contains a constructor function for creating a [Rayleigh][rayleigh-distribution] distribution object.

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<div class="namespace-toc">

-   <span class="signature">[`Rayleigh( [sigma] )`][@stdlib/stats/base/dists/rayleigh/ctor]</span><span class="delimiter">: </span><span class="description">Rayleigh distribution constructor.</span>

</div>

<!-- </toc> -->

```javascript
var Rayleigh = require( '@stdlib/stats/base/dists/rayleigh' ).Rayleigh;

var dist = new Rayleigh( 2.0 );

var y = dist.pdf( 0.8 );
// returns ~0.185
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- TODO: better examples -->

<!-- eslint no-undef: "error" -->

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

/*
* The Rayleigh distribution can be used to model wind speeds.
* Let's consider a scenario where we want to estimate various properties related to wind speeds.
*/

// Set the Rayleigh distribution parameter (scale parameter):
var s = 10.0;

// Calculate mean, variance, and standard deviation of the Rayleigh distribution:
console.log( rayleigh.mean( s ) );
// => ~12.533

console.log( rayleigh.variance( s ) );
// => ~42.920

console.log( rayleigh.stdev( s ) );
// => ~6.551

// Evaluate the Probability Density Function (PDF) for a specific wind speed:
var w = 15.0;
console.log( rayleigh.pdf( w, s ) );
// => ~0.049

// Determine Cumulative Distribution Function (CDF) for wind speeds up to a certain value:
var t = 15.0;
console.log( rayleigh.cdf( t, s ) );
// => ~0.675

// Calculate the probability of wind speeds exceeding the threshold:
var p = 1.0 - rayleigh.cdf( t, s );
console.log( 'Probability of wind speeds exceeding ' + t + ' m/s:', p );

// Find the wind speed at which there's a 70% chance it won't exceed using the Quantile function:
var c = 0.7;
console.log( rayleigh.quantile( c, s ) );
// => ~15.518
```

</section>

<!-- /.examples -->

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<section class="related">

</section>

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<section class="links">

[rayleigh-distribution]: https://en.wikipedia.org/wiki/Rayleigh_distribution

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[@stdlib/stats/base/dists/rayleigh/ctor]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/ctor

[@stdlib/stats/base/dists/rayleigh/entropy]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/entropy

[@stdlib/stats/base/dists/rayleigh/kurtosis]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/kurtosis

[@stdlib/stats/base/dists/rayleigh/mean]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/mean

[@stdlib/stats/base/dists/rayleigh/median]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/median

[@stdlib/stats/base/dists/rayleigh/mode]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/mode

[@stdlib/stats/base/dists/rayleigh/skewness]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/skewness

[@stdlib/stats/base/dists/rayleigh/stdev]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/stdev

[@stdlib/stats/base/dists/rayleigh/variance]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/variance

[@stdlib/stats/base/dists/rayleigh/cdf]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/cdf

[@stdlib/stats/base/dists/rayleigh/logcdf]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/logcdf

[@stdlib/stats/base/dists/rayleigh/logpdf]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/logpdf

[@stdlib/stats/base/dists/rayleigh/mgf]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/mgf

[@stdlib/stats/base/dists/rayleigh/pdf]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/pdf

[@stdlib/stats/base/dists/rayleigh/quantile]: https://github.com/stdlib-js/stats/tree/main/base/dists/rayleigh/quantile

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</section>

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