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# Logarithm of Cumulative Distribution Function

> [Gumbel][gumbel-distribution] distribution logarithm of [cumulative distribution function][cdf].

<section class="intro">

The [cumulative distribution function][cdf] for a [Gumbel][gumbel-distribution] random variable is

<!-- <equation class="equation" label="eq:gumbel_cdf" align="center" raw="F\left( x; \mu, \beta \right ) = e^{{-e^{{-(x-\mu )/\beta }}}}" alt="Cumulative distribution function for a Gumbel distribution."> -->

<div class="equation" align="center" data-raw-text="F\left( x; \mu, \beta \right ) = e^{{-e^{{-(x-\mu )/\beta }}}}" data-equation="eq:gumbel_cdf">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/stats/base/dists/gumbel/logcdf/docs/img/equation_gumbel_cdf.svg" alt="Cumulative distribution function for a Gumbel distribution.">
    <br>
</div>

<!-- </equation> -->

where `mu` is the location parameter and `beta > 0` is the scale parameter.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### logcdf( x, mu, beta )

Evaluates the logarithm of the [cumulative distribution function][cdf] (CDF) for a [Gumbel][gumbel-distribution] distribution with parameters `mu` (location parameter) and `beta` (scale parameter).

```javascript
var y = logcdf( 10.0, 0.0, 3.0 );
// returns ~-0.036

y = logcdf( -2.0, 0.0, 3.0 );
// returns ~-1.948

y = logcdf( 0.0, 0.0, 1.0 );
// returns ~-1
```

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

```javascript
var y = logcdf( NaN, 0.0, 1.0 );
// returns NaN

y = logcdf( 0.0, NaN, 1.0 );
// returns NaN

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

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

```javascript
var y = logcdf( 2.0, 0.0, -1.0 );
// returns NaN

y = logcdf( 2.0, 0.0, 0.0 );
// returns NaN
```

#### logcdf.factory( mu, beta )

Returns a function for evaluating the logarithm of the [cumulative distribution function][cdf] of a [Gumbel][gumbel-distribution] distribution with parameters `mu` (location parameter) and `beta` (scale parameter).

```javascript
var mylogcdf = logcdf.factory( 0.0, 3.0 );

var y = mylogcdf( 10.0 );
// returns ~-0.036

y = mylogcdf( -2.0 );
// returns ~-1.948
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   In virtually all cases, using the `logpdf` or `logcdf` functions is preferable to manually computing the logarithm of the `pdf` or `cdf`, respectively, since the latter is prone to overflow and underflow.

</section>

<!-- /.notes -->

<section class="examples">

## Examples

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

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

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

for ( i = 0; i < 100; i++ ) {
    x = randu() * 10.0;
    mu = randu() * 10.0;
    beta = randu() * 10.0;
    y = logcdf( x, mu, beta );
    console.log( 'x: %d, µ: %d, β: %d, ln(F(x;µ,β)): %d', x.toFixed( 4 ), mu.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

<section class="links">

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

[cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function

</section>

<!-- /.links -->
