npm package: @stdlib/stats-base-dists-cauchy-logpdf
Description: Cauchy distribution logarithm of probability density function (logPDF).
Author: The Stdlib Authors
Latest version: 0.2.3

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# Logarithm of Probability Density Function

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> [Cauchy][cauchy-distribution] distribution logarithm of probability density function (logPDF).

<section class="intro">

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

<!-- <equation class="equation" label="eq:cauchy_cauchy_pdf" align="center" raw="f(x;\gamma,x_0)=\frac{1}{\pi\gamma\,\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}\!" alt="Probability density function (PDF) for a Cauchy distribution."> -->

<div class="equation" align="center" data-raw-text="f(x;\gamma,x_0)=\frac{1}{\pi\gamma\,\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}\!" data-equation="eq:cauchy_cauchy_pdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/cauchy/logpdf/docs/img/equation_cauchy_cauchy_pdf.svg" alt="Probability density function (PDF) for a Cauchy distribution.">
    <br>
</div>

<!-- </equation> -->

where `x0` is the location parameter and `gamma > 0` is the scale parameter.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-cauchy-logpdf
```

</section>

<section class="usage">

## Usage

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

#### logpdf( x, x0, gamma )

Evaluates the natural logarithm of the [probability density function][pdf] (PDF) for a [Cauchy][cauchy-distribution] distribution with parameters `x0` (location parameter) and `gamma > 0` (scale parameter).

```javascript
var y = logpdf( 2.0, 1.0, 1.0 );
// returns ~-1.838

y = logpdf( 4.0, 3.0, 0.1 );
// returns ~-3.457

y = logpdf( 4.0, 3.0, 3.0 );
// returns ~-2.349
```

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

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

y = logpdf( 2.0, NaN, 1.0 );
// returns NaN

y = logpdf( 2.0, 1.0, NaN );
// returns NaN
```

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

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

#### logpdf.factory( x0, gamma )

Returns a `function` for evaluating the natural logarithm of the [PDF][pdf] of a [Cauchy][cauchy-distribution] distribution with location parameter `x0` and scale parameter `gamma`.

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

var y = mylogpdf( 10.0 );
// returns ~-1.838

y = mylogpdf( 5.0 );
// returns ~-3.819
```

</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 uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var EPS = require( '@stdlib/constants-float64-eps' );
var logpdf = require( '@stdlib/stats-base-dists-cauchy-logpdf' );

var opts = {
    'dtype': 'float64'
};
var gamma = uniform( 10, EPS, 20.0, opts );
var x0 = uniform( 10, -5.0, 5.0, opts );
var x = uniform( 10, 0.0, 10.0, opts );

logEachMap( 'x: %0.4f, x0: %0.4f, γ: %0.4f, ln(f(x;x0,γ)): %0.4f', x, x0, gamma, logpdf );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/cauchy/logpdf.h"
```

#### stdlib_base_dists_cauchy_logpdf( x, x0, gamma )

Evaluates the natural logarithm of the [probability density function][pdf] (PDF) for a [Cauchy][cauchy-distribution] distribution with parameters `x0` (location parameter) and `gamma > 0` (scale parameter).

```c
double out = stdlib_base_dists_cauchy_logpdf( 2.0, 1.0, 1.0 );
// returns ~-1.838
```

The function accepts the following arguments:

-   **x**: `[in] double` input value.
-   **x0**: `[in] double` location parameter.
-   **gamma**: `[in] double` scale parameter.

```c
double stdlib_base_dists_cauchy_logpdf( const double x, const double x0, const double gamma );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/cauchy/logpdf.h"
#include "stdlib/constants/float64/eps.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v*(max-min) );
}

int main( void ) {
    double gamma;
    double x0;
    double x;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        x = random_uniform( 0.0, 10.0 );
        x0 = random_uniform( -5.0, 5.0 );
        gamma = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 20.0 );
        y = stdlib_base_dists_cauchy_logpdf( x, x0, gamma );
        printf( "x: %lf, x0: %lf, γ: %lf, ln(f(x;x0,γ)): %lf\n", x, x0, gamma, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

</section>

<!-- /.related -->

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<section class="main-repo" >

* * *

## Notice

This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

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---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-base-dists-cauchy-logpdf/main/LICENSE

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

[cauchy-distribution]: https://en.wikipedia.org/wiki/Cauchy_distribution

</section>

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