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

Standard library statistical functions.

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# Probability Density Function

> [Student's t][t-distribution] distribution probability density function (PDF).

<section class="intro">

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

<!-- <equation class="equation" label="eq:t_pdf" align="center" raw="\frac{1} {\sqrt{\nu}\,B\left( \tfrac{1}{2}, \tfrac{\nu}{2} \right )} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}" alt="Probability density function (PDF) for a Student's t distribution."> -->

<div class="equation" align="center" data-raw-text="\frac{1} {\sqrt{\nu}\,B\left( \tfrac{1}{2}, \tfrac{\nu}{2} \right )} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}" data-equation="eq:t_pdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/t/pdf/docs/img/equation_t_pdf.svg" alt="Probability density function (PDF) for a Student's t distribution.">
    <br>
</div>

<!-- </equation> -->

where `v > 0` is the degrees of freedom.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var pdf = require( '@stdlib/stats/base/dists/t/pdf' );
```

#### pdf( x, v )

Evaluates the [probability density function][pdf] (PDF) for a [Student's t][t-distribution] distribution with degrees of freedom `v`.

```javascript
var y = pdf( 0.3, 4.0 );
// returns ~0.355

y = pdf( 2.0, 0.7 );
// returns ~0.058

y = pdf( -1.0, 0.5 );
// returns ~0.118
```

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

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

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

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

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

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

#### pdf.factory( v )

Returns a `function` for evaluating the [PDF][pdf] of a [Student's t][t-distribution] distribution with degrees of freedom `v`.

```javascript
var mypdf = pdf.factory( 1.0 );
var y = mypdf( 3.0 );
// returns ~0.032

y = mypdf( 1.0 );
// returns ~0.159
```

</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/t/pdf' );

var v;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = (randu() * 6.0) - 3.0;
    v = randu() * 10.0;
    y = pdf( x, v );
    console.log( 'x: %d, v: %d, f(x;v): %d', x, v, y );
}
```

</section>

<!-- /.examples -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

</section>

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<section class="links">

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

[t-distribution]: https://en.wikipedia.org/wiki/Student%27s_t-distribution

</section>

<!-- /.links -->