<!--

@license Apache-2.0

Copyright (c) 2018 The Stdlib Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

   http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.

-->

# Logarithm of Probability Density Function

> [Degenerate distribution][degenerate-distribution] logarithm of [probability density function][pdf] (logPDF).

<section class="intro">

Strictly speaking, as a discrete distribution, a [degenerate][degenerate-distribution] has no [probability density function][pdf] (PDF). Extending the notion of a [PDF][pdf], we conceptualize the [PDF][pdf] of a [degenerate][degenerate-distribution] distribution as an infinitely tall spike centered at `mu`. More formally,

<!-- <equation class="equation" label="eq:degenerate_pdf" align="center" raw="f(x;\mu) = \delta(x-\mu)" alt="Probability density function (PDF) for a degenerate distribution."> -->

<div class="equation" align="center" data-raw-text="f(x;\mu) = \delta(x-\mu)" data-equation="eq:degenerate_pdf">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/9488a06f69a99a22b91882d8b44881fc807708eb/lib/node_modules/@stdlib/stats/base/dists/degenerate/logpdf/docs/img/equation_degenerate_pdf.svg" alt="Probability density function (PDF) for a degenerate distribution.">
    <br>
</div>

<!-- </equation> -->

where `delta` is the Dirac delta function.

<!-- <equation class="equation" label="eq:dirac_delta" align="center" raw="\delta(x) = {\begin{cases}+\infty , & x = 0\\0, & x \neq 0\end{cases}}" alt="Dirac delta function."> -->

<div class="equation" align="center" data-raw-text="\delta(x) = {\begin{cases}+\infty , &amp; x = 0\\0, &amp; x \neq 0\end{cases}}" data-equation="eq:dirac_delta">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/9488a06f69a99a22b91882d8b44881fc807708eb/lib/node_modules/@stdlib/stats/base/dists/degenerate/logpdf/docs/img/equation_dirac_delta.svg" alt="Dirac delta function.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### logpdf( x, mu )

Evaluates the natural logarithm of the [PDF][pdf] of a [degenerate distribution][degenerate-distribution] centered at `mu`.

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

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

#### logpdf.factory( mu )

Returns a function for evaluating the natural logarithm of the [PDF][pdf] of a [degenerate distribution][degenerate-distribution] centered at `mu`.

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

var y = mylogpdf( 10.0 );
// returns Infinity

y = mylogpdf( 5.0 );
// returns -Infinity

y = mylogpdf( 12.0 );
// returns -Infinity
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var logpdf = require( '@stdlib/stats/base/dists/degenerate/logpdf' );

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

for ( i = 0; i < 100; i++ ) {
    x = round( randu()*5.0 );
    mu = round( randu()*5.0 );
    y = logpdf( x, mu );
    console.log( 'x: %d, µ: %d, ln(f(x;µ)): %d', x, mu, y );
}
```

</section>

<!-- /.examples -->

<section class="links">

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

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

</section>

<!-- /.links -->
