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# Logarithm of Probability Density Function

> Evaluate the natural logarithm of the [probability density function][pdf] (PDF) for a [chi-squared][chisquare-distribution] distribution.

<section class="intro">

The [probability density function][pdf] (PDF) for a [chi-squared][chisquare-distribution] random variable is

<!-- <equation class="equation" label="eq:chisquare_pdf" align="center" raw="{\frac{1}{2^{\frac {k}{2}}\Gamma\left({\frac{k}{2}}\right)}}\;x^{{\frac {k}{2}}-1}e^{-{\frac {x}{2}}}" alt="Probability density function (PDF) for a chi-squared distribution."> -->

<div class="equation" align="center" data-raw-text="{\frac{1}{2^{\frac {k}{2}}\Gamma\left({\frac{k}{2}}\right)}}\;x^{{\frac {k}{2}}-1}e^{-{\frac {x}{2}}}" data-equation="eq:chisquare_pdf">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/stats/base/dists/chisquare/logpdf/docs/img/equation_chisquare_pdf.svg" alt="Probability density function (PDF) for a chi-squared distribution.">
    <br>
</div>

<!-- </equation> -->

where `k` is the degrees of freedom and `Γ` denotes the [gamma function][gamma-function].

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### logpdf( x, k )

Evaluates the natural logarithm of the [probability density function][pdf] (PDF) for a [chi-squared][chisquare-distribution] distribution with degrees of freedom `k`.

```javascript
var y = logpdf( 0.1, 1.0 );
// returns ~0.182

y = logpdf( 0.5, 2.0 );
// returns ~-0.943

y = logpdf( -1.0, 4.0 );
// returns -Infinity
```

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

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

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

If provided `k < 0`, the function returns `NaN`.

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

If provided `k = 0`, the function evaluates the [PDF][pdf] of a [degenerate distribution][degenerate-distribution] centered at `0`.

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

y = logpdf( 0.0, 0.0 );
// returns Infinity
```

#### logpdf.factory( k )

Returns a `function` for evaluating the [PDF][pdf] for a [chi-squared][chisquare-distribution] distribution with degrees of freedom `k`.

```javascript
var myLogPDF = logpdf.factory( 6.0 );

var y = myLogPDF( 3.0 );
// returns ~-2.075

y = myLogPDF( 1.0 );
// returns ~-3.273
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random/base/randu' );
var logpdf = require( '@stdlib/stats/base/dists/chisquare/logpdf' );

var k;
var x;
var y;
var i;

for ( i = 0; i < 20; i++ ) {
    x = randu() * 10.0;
    k = randu() * 10.0;
    y = logpdf( x, k );
    console.log( 'x: %d, k: %d, ln(f(x;k)): %d', x.toFixed( 4 ), k.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

<section class="links">

[chisquare-distribution]: https://en.wikipedia.org/wiki/Chi-squared_distribution

[degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution

[gamma-function]: https://en.wikipedia.org/wiki/Gamma_function

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

</section>

<!-- /.links -->
