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

Standard library statistical functions.

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# Gumbel

> Gumbel distribution.

<section class="usage">

## Usage

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

#### gumbel

Gumbel distribution.

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

The namespace contains the following distribution functions:

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

<div class="namespace-toc">

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

</div>

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The namespace contains the following functions for calculating distribution properties:

<!-- <toc pattern="*+(entropy|kurtosis|mean|median|mode|skewness|stdev|variance)*"> -->

<div class="namespace-toc">

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

</div>

<!-- </toc> -->

The namespace contains a constructor function for creating a [Gumbel][gumbel-distribution] distribution object.

<!-- <toc pattern="*ctor*"> -->

<div class="namespace-toc">

-   <span class="signature">[`Gumbel( [mu, beta] )`][@stdlib/stats/base/dists/gumbel/ctor]</span><span class="delimiter">: </span><span class="description">Gumbel distribution constructor.</span>

</div>

<!-- </toc> -->

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

var dist = new Gumbel( 2.0, 4.0 );

var y = dist.pdf( 2.0 );
// returns ~0.092
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- TODO: better examples -->

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

```javascript
var Float64Array = require( '@stdlib/array/float64' );
var filledarrayBy = require( '@stdlib/array/filled-by' );
var mean = require( '@stdlib/stats/base/mean' );
var variance = require( '@stdlib/stats/base/variance' );
var stdev = require( '@stdlib/stats/base/stdev' );
var randGumbel = require( '@stdlib/random/base/gumbel' ).factory;
var gumbel = require( '@stdlib/stats/base/dists/gumbel' );

// Set the parameters of the Gumbel distribution:
var mu = 30.0;   // Location parameter (e.g., average annual maximum temperature in °C)
var beta = 5.0;  // Scale parameter

// Simulate annual maximum daily temperatures over 1000 years:
var N = 1000;
var rgumbel = randGumbel( mu, beta );
var maxTemperatures = filledarrayBy( N, 'float64', rgumbel );

// Compute theoretical statistics of the Gumbel distribution:
var theoreticalMean = gumbel.mean( mu, beta);
var theoreticalVariance = gumbel.variance( mu, beta );
var theoreticalStdev = gumbel.stdev( mu, beta );

// Compute sample statistics of the simulated data:
var sampleMean = mean( N, maxTemperatures, 1 );
var sampleVariance = variance( N, 1, maxTemperatures, 1 ); // with Bessel's correction
var sampleStdev = stdev( N, 1, maxTemperatures, 1 ); // with Bessel's correction

// Display theoretical and sample statistics:
console.log( '--- Statistical Comparison ---\n' );
console.log( 'Mean:');
console.log( '  Theoretical: %d°C', theoreticalMean.toFixed(2) );
console.log( '  Sample:      %d°C\n', sampleMean.toFixed(2) );
console.log( 'Variance:');
console.log( '  Theoretical: %d°C²', theoreticalVariance.toFixed(2) );
console.log( '  Sample:      %d°C²\n', sampleVariance.toFixed(2) );
console.log( 'Standard Deviation:' );
console.log( '  Theoretical: %d°C', theoreticalStdev.toFixed(2) );
console.log( '  Sample:      %d°C\n', sampleStdev.toFixed(2) );

// Define quantile probabilities:
var p = new Float64Array( [ 0.25, 0.5, 0.75 ] );
var label = [ 'First Quartile', 'Median', 'Third Quartile' ];
var theoreticalQuantiles = new Float64Array([
    gumbel.quantile( p[0], mu, beta ),
    gumbel.quantile( p[1], mu, beta ),
    gumbel.quantile( p[2], mu, beta )
]);

console.log( 'Quantiles:' );
var i;
for ( i = 0; i < p.length; i++ ) {
    console.log( label[i] + ': %d°C', theoreticalQuantiles[i].toFixed(2) );
}
```

</section>

<!-- /.examples -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

</section>

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<section class="links">

[gumbel-distribution]: https://en.wikipedia.org/wiki/Gumbel_distribution

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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</section>

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