@stdlib/stats/base/dists/logistic/logpdf/README.md

Standard library statistical functions.

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# Logarithm of Probability Density Function

> [Logistic][logistic-distribution] distribution logarithm of [probability density function (PDF)][pdf].

<section class="intro">

The [probability density function][pdf] (PDF) for a [logistic][logistic-distribution] random variable is

<!-- <equation class="equation" label="eq:logistic_pdf" align="center" raw="f(x; \mu,s) = \frac{e^{-\frac{x-\mu}{s}}} {s\left(1+e^{-\frac{x-\mu}{s}}\right)^2}" alt="Probability density function (PDF) for a logistic distribution."> -->

<div class="equation" align="center" data-raw-text="f(x; \mu,s) = \frac{e^{-\frac{x-\mu}{s}}} {s\left(1+e^{-\frac{x-\mu}{s}}\right)^2}" data-equation="eq:logistic_pdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/logistic/logpdf/docs/img/equation_logistic_pdf.svg" alt="Probability density function (PDF) for a logistic distribution.">
    <br>
</div>

<!-- </equation> -->

where `mu` is the location parameter and `s` is the scale parameter.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### logpdf( x, mu, s )

Evaluates the logarithm of the [probability density function][pdf] (PDF) for a [logistic][logistic-distribution] distribution with parameters `mu` (location parameter) and `s` (scale parameter).

```javascript
var y = logpdf( 2.0, 0.0, 1.0 );
// returns ~-2.254

y = logpdf( -1.0, 4.0, 4.0 );
// returns ~-3.14
```

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

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

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

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

If provided `s < 0`, the function returns `NaN`.

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

If provided `s = 0`, the function evaluates the logarithm of the [PDF][pdf] of a [degenerate distribution][degenerate-distribution] centered at `mu`.

```javascript
var y = logpdf( 2.0, 8.0, 0.0 );
// returns -Infinity

y = logpdf( 8.0, 8.0, 0.0 );
// returns Infinity
```

#### logpdf.factory( mu, s )

Returns a function for evaluating the logarithm of the [probability density function][pdf] (PDF) of a [logistic][logistic-distribution] distribution with parameters `mu` (location parameter) and `s` (scale parameter).

```javascript
var mylogpdf = logpdf.factory( 10.0, 2.0 );

var y = mylogpdf( 10.0 );
// returns ~-2.079

y = mylogpdf( 5.0 );
// returns ~-3.351
```

</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 logpdf = require( '@stdlib/stats/base/dists/logistic/logpdf' );

var mu;
var s;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    mu = randu() * 10.0;
    s = randu() * 10.0;
    y = logpdf( x, mu, s );
    console.log( 'x: %d, µ: %d, s: %d, ln(f(x;µ,s)): %d', x, mu, s, y );
}
```

</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">

[logistic-distribution]: https://en.wikipedia.org/wiki/Logistic_distribution

[pdf]: https://en.wikipedia.org/wiki/Probability_density_function

[degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution

</section>

<!-- /.links -->