npm package: @stdlib/stats-base-dists-t-cdf
Description: Student's t distribution cumulative distribution function (CDF).
Author: The Stdlib Authors
Latest version: 0.2.3

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# Cumulative Distribution Function

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> [Student's t][t-distribution] distribution [cumulative distribution function][cdf] (CDF).

<section class="intro">

The [cumulative distribution function][cdf] (CDF) for a [t distribution][t-distribution] random variable is

<!-- <equation class="equation" label="eq:t_cdf" align="center" raw="F(x;\nu) = 1 - \frac{1}{2} \frac{\operatorname{Beta}(\tfrac{\nu}{\nu + x^2};\,\tfrac{\nu}{2},\tfrac{1}{2})}{\operatorname{Beta}(\tfrac{\nu}{2}, \tfrac{1}{2})}" alt="Cumulative distribution function (CDF) for a Student's t distribution."> -->

<div class="equation" align="center" data-raw-text="F(x;\nu) = 1 - \frac{1}{2} \frac{\operatorname{Beta}(\tfrac{\nu}{\nu + x^2};\,\tfrac{\nu}{2},\tfrac{1}{2})}{\operatorname{Beta}(\tfrac{\nu}{2}, \tfrac{1}{2})}" data-equation="eq:t_cdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/t/cdf/docs/img/equation_t_cdf.svg" alt="Cumulative distribution function (CDF) for a Student's t distribution.">
    <br>
</div>

<!-- </equation> -->

where `v > 0` is the degrees of freedom. In the definition, `Beta( x; a, b )` denotes the lower incomplete beta function and `Beta( a, b )` the [beta function][beta-function].

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-t-cdf
```

</section>

<section class="usage">

## Usage

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

#### cdf( x, v )

Evaluates the [cumulative distribution function][cdf] (CDF) for a [Student's t][t-distribution] distribution with degrees of freedom `v`.

```javascript
var y = cdf( 2.0, 0.1 );
// returns ~0.611

y = cdf( 1.0, 2.0 );
// returns ~0.789

y = cdf( -1.0, 4.0 );
// returns ~0.187
```

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

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

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

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

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

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

#### cdf.factory( v )

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

```javascript
var mycdf = cdf.factory( 0.5 );
var y = mycdf( 3.0 );
// returns ~0.816

y = mycdf( 1.0 );
// returns ~0.699
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

```javascript
var uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var cdf = require( '@stdlib/stats-base-dists-t-cdf' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 10, -3.0, 3.0, opts );
var v = uniform( 10, 0.0, 10.0, opts );

logEachMap( 'x: %0.4f, v: %0.4f, F(x;v): %0.4f', x, v, cdf );
```

</section>

<!-- /.examples -->

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

<section class="related">

</section>

<!-- /.related -->

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<section class="main-repo" >

* * *

## Notice

This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

#### Community

[![Chat][chat-image]][chat-url]

---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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[beta-function]: https://en.wikipedia.org/wiki/Beta_function

[cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function

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

</section>

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