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# Entropy

> [Gumbel][gumbel-distribution] distribution [differential entropy][entropy].

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The [differential entropy][entropy] for a [gumbel][gumbel-distribution] random variable with location `μ` and scale `β` is

<!-- <equation class="equation" label="eq:gumbel_entropy" align="center" raw="h\left( X \right) = \ln(\beta )+\gamma+1" alt="Differential entropy for a gumbel distribution."> -->

<div class="equation" align="center" data-raw-text="h\left( X \right) = \ln(\beta )+\gamma+1" data-equation="eq:gumbel_entropy">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/gumbel/entropy/docs/img/equation_gumbel_entropy.svg" alt="Differential entropy for a gumbel distribution.">
    <br>
</div>

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where `γ` is the [Euler–Mascheroni constant][euler-mascheroni].

</section>

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<section class="usage">

## Usage

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

#### entropy( mu, beta )

Returns the [differential entropy][entropy] for a [Gumbel][gumbel-distribution] distribution with location parameter `mu` and scale parameter `beta` (in [nats][nats]).

```javascript
var y = entropy( 2.0, 1.0 );
// returns ~1.577

y = entropy( 0.0, 1.0 );
// returns ~1.577

y = entropy( -1.0, 4.0 );
// returns ~2.964
```

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

```javascript
var y = entropy( NaN, 1.0 );
// returns NaN

y = entropy( 0.0, NaN );
// returns NaN
```

If provided `beta <= 0`, the function returns `NaN`.

```javascript
var y = entropy( 0.0, 0.0 );
// returns NaN

y = entropy( 0.0, -1.0 );
// returns NaN
```

</section>

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</section>

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<section class="examples">

## Examples

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

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

var beta;
var mu;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    mu = ( randu()*10.0 ) - 5.0;
    beta = randu() * 20.0;
    y = entropy( mu, beta );
    console.log( 'µ: %d, β: %d, h(X;µ,β): %d', mu.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</section>

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</section>

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<section class="related">

</section>

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<section class="links">

[euler-mascheroni]: https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant

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

[entropy]: https://en.wikipedia.org/wiki/Entropy_%28information_theory%29

[nats]: https://en.wikipedia.org/wiki/Nat_%28unit%29

</section>

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