@stdlib/stats
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Standard library statistical functions.
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# Probability Density Function
> [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}} & \text{ for }x\ge \beta \\ 0 & \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="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 randu = require( '@stdlib/random/base/randu' );
var pdf = require( '@stdlib/stats/base/dists/pareto-type1/pdf' );
var alpha;
var beta;
var x;
var y;
var i;
for ( i = 0; i < 10; i++ ) {
x = randu() * 8.0;
alpha = randu() * 4.0;
beta = randu() * 4.0;
y = pdf( x, alpha, beta );
console.log( 'x: %d, α: %d, β: %d, f(x;α,β): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}
```
</section>
<!-- /.examples -->
<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->
<section class="related">
</section>
<!-- /.related -->
<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
<section class="links">
[pareto-distribution]: https://en.wikipedia.org/wiki/Pareto_distribution
[pdf]: https://en.wikipedia.org/wiki/Probability_density_function
</section>
<!-- /.links -->