@stdlib/stats/base/dists/levy/logcdf/README.md

Standard library statistical functions.

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# Logarithm of Cumulative Distribution Function

> [Lévy][levy-distribution] distribution logarithm of [cumulative distribution function][cdf].

<section class="intro">

The [cumulative distribution function][cdf] for a [Lévy][levy-distribution] random variable is

<!-- <equation class="equation" label="eq:levy_cdf" align="center" raw="F(x;\mu,b) = \begin{cases} \operatorname{erfc}\left(\sqrt{\frac{c}{2(x-\mu)}}\right) & \text{ for } x > \mu \\ 0 & \text{ otherwise } \end{cases}" alt="Cumulative distribution function for a Lévy distribution."> -->

<div class="equation" align="center" data-raw-text="F(x;\mu,b) = \begin{cases} \operatorname{erfc}\left(\sqrt{\frac{c}{2(x-\mu)}}\right) &amp; \text{ for } x &gt; \mu \\ 0 &amp; \text{ otherwise } \end{cases}" data-equation="eq:levy_cdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/levy/logcdf/docs/img/equation_levy_cdf.svg" alt="Cumulative distribution function for a Lévy distribution.">
    <br>
</div>

<!-- </equation> -->

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

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### logcdf( x, mu, c )

Evaluates the logarithm of the [cumulative distribution function][cdf] (CDF) for a [Lévy][levy-distribution] distribution with parameters `mu` (location parameter) and `c > 0` (scale parameter).

```javascript
var y = logcdf( 2.0, 0.0, 1.0 );
// returns ~-0.735

y = logcdf( 12.0, 10.0, 3.0 );
// returns ~-1.51

y = logcdf( 9.0, 10.0, 3.0 );
// returns -Infinity
```

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 `c <= 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, c )

Returns a function for evaluating the logarithm of the [cumulative distribution function][cdf] of a [Lévy][levy-distribution] distribution with parameters `mu` (location parameter) and `c > 0` (scale parameter).

```javascript
var mylogcdf = logcdf.factory( 3.0, 1.5 );

var y = mylogcdf( 4.0 );
// returns ~-1.511

y = mylogcdf( 2.0 );
// returns -Infinity
```

</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 EPS = require( '@stdlib/constants/float64/eps' );
var logcdf = require( '@stdlib/stats/base/dists/levy/logcdf' );

var mu;
var c;
var x;
var y;
var i;

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

</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">

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

[levy-distribution]: https://en.wikipedia.org/wiki/L%C3%A9vy_distribution

</section>

<!-- /.links -->