@stdlib/stats/base/dists/normal/logpdf/README.md

Standard library statistical functions.

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# Logarithm of Probability Density Function

> Evaluate the natural logarithm of the probability density function (PDF) for a [normal][normal-distribution] distribution.

<section class="intro">

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

<!-- <equation class="equation" label="eq:normal_normal_pdf" align="center" raw="f(x;\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2 \sigma^2}}" alt="Probability density function (PDF) for a normal distribution."> -->

<div class="equation" align="center" data-raw-text="f(x;\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2 \sigma^2}}" data-equation="eq:normal_normal_pdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/normal/logpdf/docs/img/equation_normal_normal_pdf.svg" alt="Probability density function (PDF) for a normal distribution.">
    <br>
</div>

<!-- </equation> -->

where `µ` is the mean and `σ` is the standard deviation.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### logpdf( x, mu, sigma )

Evaluates the natural logarithm of the [probability density function][pdf] (PDF) for a [normal][normal-distribution] distribution with parameters `mu` (mean) and `sigma` (standard deviation).

```javascript
var y = logpdf( 2.0, 0.0, 1.0 );
// returns ~-2.919

y = logpdf( -1.0, 4.0, 2.0 );
// returns ~-4.737
```

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

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

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

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

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

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

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

```javascript
var y = logpdf( 2.0, 8.0, 0.0 );
// returns -Infinity

y = logpdf( 8.0, 8.0, 0.0 );
// returns Infinity
```

#### logpdf.factory( mu, sigma )

Returns a `function` for evaluating the [probability density function][pdf] (PDF) of a [normal][normal-distribution] distribution with parameters `mu` (mean) and `sigma` (standard deviation).

```javascript
var mylogpdf = logpdf.factory( 10.0, 2.0 );

var y = mylogpdf( 10.0 );
// returns ~-1.612

y = mylogpdf( 5.0 );
// returns ~-4.737
```

</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/normal/logpdf' );

var sigma;
var mu;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    mu = (randu() * 10.0) - 5.0;
    sigma = randu() * 20.0;
    y = logpdf( x, mu, sigma );
    console.log( 'x: %d, µ: %d, σ: %d, ln(f(x;µ,σ)): %d', x, mu, sigma, y );
}
```

</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">

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

[normal-distribution]: https://en.wikipedia.org/wiki/Normal_distribution

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

</section>

<!-- /.links -->