npm package: @stdlib/stats-base-dists-arcsine-pdf
Description: Arcsine distribution probability density function (PDF).
Author: The Stdlib Authors
Latest version: 0.3.1

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# 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] -->

> [Arcsine][arcsine-distribution] distribution [probability density function][pdf] (PDF).

<section class="intro">

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

<!-- <equation class="equation" label="eq:arcsine_pdf" align="center" raw="f(x;a,b)=\begin{cases} {\frac{1}{\pi {\sqrt{(x-a)(b-x)}}}} & \text{for } x \in [a,b] \\ 0 & \text{otherwise} \end{cases}" alt="Probability density function (PDF) for an arcsine distribution."> -->

<div class="equation" align="center" data-raw-text="f(x;a,b)=\begin{cases} {\frac{1}{\pi {\sqrt{(x-a)(b-x)}}}} &amp; \text{for } x \in [a,b] \\ 0 &amp; \text{otherwise} \end{cases}" data-equation="eq:arcsine_pdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/arcsine/pdf/docs/img/equation_arcsine_pdf.svg" alt="Probability density function (PDF) for an arcsine distribution.">
    <br>
</div>

<!-- </equation> -->

where `a` is the minimum support and `b` is the maximum support of the distribution. The parameters must satisfy `a < b`.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-arcsine-pdf
```

</section>

<section class="usage">

## Usage

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

#### pdf( x, a, b )

Evaluates the [probability density function][pdf] (PDF) for an [arcsine][arcsine-distribution] distribution with parameters `a` (minimum support) and `b` (maximum support).

```javascript
var y = pdf( 2.0, 0.0, 4.0 );
// returns ~0.159

y = pdf( 5.0, 0.0, 4.0 );
// returns 0.0

y = pdf( 0.25, 0.0, 1.0 );
// returns ~0.735
```

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

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

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

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

If provided `a >= b`, the function returns `NaN`.

```javascript
var y = pdf( 2.5, 3.0, 2.0 );
// returns NaN

y = pdf( 2.5, 3.0, 3.0 );
// returns NaN
```

#### pdf.factory( a, b )

Returns a `function` for evaluating the [PDF][pdf] of an [arcsine][arcsine-distribution] distribution with parameters `a` (minimum support) and `b` (maximum support).

```javascript
var myPDF = pdf.factory( 6.0, 7.0 );
var y = myPDF( 7.0 );
// returns Infinity

y = myPDF( 5.0 );
// returns 0.0
```

</section>

<!-- /.usage -->

<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 pdf = require( '@stdlib/stats-base-dists-arcsine-pdf' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 25, -10.0, 10.0, opts );
var a = uniform( x.length, -20.0, 0.0, opts );
var b = uniform( x.length, 0.0, 40.0, opts );

logEachMap( 'x: %0.4f, a: %0.4f, b: %0.4f, f(x;a,b): %0.4f', x, a, b, pdf );
```

</section>

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

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<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/arcsine/pdf.h"
```

#### stdlib_base_dists_arcsine_pdf( x, a, b )

Evaluates the probability density function (PDF) for an arcsine distribution.

```c
double out = stdlib_base_dists_arcsine_pdf( 0.5, 0.0, 4.0 );
// returns ~0.159
```

The function accepts the following arguments:

-   **x**: `[in] double` input value.
-   **a**: `[in] double` minimum support.
-   **b**: `[in] double` maximum support.

```c
double stdlib_base_dists_arcsine_pdf( const double x, const double a, const double b );
```

</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/arcsine/pdf.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 x;
    double a;
    double b;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        x = random_uniform( -10.0, 10.0 );
        a = random_uniform( -20.0, 0.0 );
        b = random_uniform( a, a+40.0 );
        y = stdlib_base_dists_arcsine_pdf( x, a, b );
        printf( "x: %lf, a: %lf, b: %lf, f(x;a,b): %lf\n", x, a, b, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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<section class="references">

</section>

<!-- /.references -->

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<section class="related">

</section>

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

[arcsine-distribution]: https://en.wikipedia.org/wiki/Arcsine_distribution

</section>

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