@stdlib/stats/base/dists/invgamma/pdf/README.md

Standard library statistical functions.

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# Probability Density Function

> [Inverse gamma][inverse-gamma] distribution probability density function (PDF).

<section class="intro">

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

<!-- <equation class="equation" label="eq:invgamma_pdf" align="center" raw="{\displaystyle f(x;\alpha ,\beta )={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{-\alpha -1}\exp \left(-{\frac {\beta }{x}}\right)}" alt="Probability density function (PDF) for an inverse gamma distribution."> -->

<div class="equation" align="center" data-raw-text="{\displaystyle f(x;\alpha ,\beta )={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{-\alpha -1}\exp \left(-{\frac {\beta }{x}}\right)}" data-equation="eq:invgamma_pdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/invgamma/pdf/docs/img/equation_invgamma_pdf.svg" alt="Probability density function (PDF) for an inverse gamma 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/invgamma/pdf' );
```

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

Evaluates the [probability density function][pdf] (PDF) for an [inverse gamma][inverse-gamma]  distribution with parameters `alpha` (shape parameter) and `beta` (rate parameter).

```javascript
var y = pdf( 2.0, 0.5, 1.0 );
// returns ~0.121

y = pdf( 0.2, 1.0, 1.0 );
// returns ~0.168

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

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

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

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

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

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

Returns a `function` for evaluating the [PDF][pdf] of an [inverse gamma][inverse-gamma] distribution with parameters `alpha` (shape parameter) and `beta` (rate parameter).

```javascript
var myPDF = pdf.factory( 6.0, 7.0 );

var y = myPDF( 2.0 );
// returns ~0.231
```

</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/invgamma/pdf' );

var alpha;
var beta;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 2.0;
    alpha = randu() * 5.0;
    beta = randu() * 5.0;
    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">

[inverse-gamma]: https://en.wikipedia.org/wiki/Inverse-gamma_distribution

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

</section>

<!-- /.links -->