npm package: @stdlib/stats-base-dists-pareto-type1-pdf
Description: Pareto  distribution (Type I) 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] -->

> [Pareto (Type I)][pareto-distribution] distribution probability density function (PDF).

<section class="intro">

The [probability density function][pdf] (PDF) for a [Pareto (Type I)][pareto-distribution] random variable is

<!-- <equation class="equation" label="eq:pareto_type1_pdf" align="center" raw="f(x;\alpha,\beta) = \begin{cases} \frac{\alpha\,\beta^\alpha}{x^{\alpha+1}} & \text{ for }x\ge \beta \\ 0 & \text{otherwise} \end{cases}" alt="Probability density function (PDF) for a Pareto (Type I) distribution."> -->

<div class="equation" align="center" data-raw-text="f(x;\alpha,\beta) = \begin{cases} \frac{\alpha\,\beta^\alpha}{x^{\alpha+1}} &amp; \text{ for }x\ge \beta \\ 0 &amp; \text{otherwise} \end{cases}" data-equation="eq:pareto_type1_pdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/pareto-type1/pdf/docs/img/equation_pareto_type1_pdf.svg" alt="Probability density function (PDF) for a Pareto (Type I) distribution.">
    <br>
</div>

<!-- </equation> -->

where `alpha > 0` is the shape parameter and `beta > 0` is the scale parameter.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-pareto-type1-pdf
```

</section>

<section class="usage">

## Usage

```javascript
var pdf = require( '@stdlib/stats-base-dists-pareto-type1-pdf' );
```

#### pdf( x, alpha, beta )

Evaluates the [probability density function][pdf] (PDF) for a [Pareto (Type I)][pareto-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (scale parameter).

```javascript
var y = pdf( 4.0, 1.0, 1.0 );
// returns ~0.063

y = pdf( 20.0, 1.0, 10.0 );
// returns 0.025

y = pdf( 7.0, 2.0, 6.0 );
// returns ~0.21

y = pdf( 7.0, 6.0, 3.0 );
// returns ~0.005

y = pdf( 1.0, 4.0, 2.0 );
// returns 0.0

y = pdf( 1.5, 4.0, 2.0 );
// returns 0.0
```

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

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

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

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

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

```javascript
var y = pdf( 2.0, -1.0, 0.5 );
// returns NaN

y = pdf( 2.0, 0.0, 0.5 );
// returns NaN
```

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

```javascript
var y = pdf( 2.0, 0.5, -1.0 );
// returns NaN

y = pdf( 2.0, 0.5, 0.0 );
// returns NaN
```

#### pdf.factory( alpha, beta )

Returns a function for evaluating the [probability density function][pdf] (PDF) (CDF) of a [Pareto (Type I)][pareto-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (scale parameter).

```javascript
var mypdf = pdf.factory( 0.5, 0.5 );
var y = mypdf( 0.8 );
// returns ~0.494

y = mypdf( 2.0 );
// returns ~0.125
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

```javascript
var uniform = require( '@stdlib/random-array-uniform' );
var EPS = require( '@stdlib/constants-float64-eps' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var pdf = require( '@stdlib/stats-base-dists-pareto-type1-pdf' );

var opts = {
    'dtype': 'float64'
};
var alpha = uniform( 10, EPS, 5.0, opts );
var beta = uniform( 10, 1.0, 5.0, opts );
var x = uniform( 10, 2.0, 15.0, opts );

logEachMap( 'x: %0.4f, α: %0.4f, β: %0.4f, f(x;α,β): %0.4f', x, alpha, beta, pdf );
```

</section>

<!-- /.examples -->

<!-- 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].

#### Community

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---

## License

See [LICENSE][stdlib-license].


## Copyright

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

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[pareto-distribution]: https://en.wikipedia.org/wiki/Pareto_distribution

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

</section>

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