npm package: @stdlib/stats-base-dists-t-entropy
Description: Student's t distribution entropy.
Author: The Stdlib Authors
Latest version: 0.3.1

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# Entropy

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> [Student's t][t-distribution] distribution [differential entropy][entropy].

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<section class="intro">

The [differential entropy][entropy] (in [nats][nats]) for a [Student's t][t-distribution] random variable with degrees of freedom `ν` is

<!-- <equation class="equation" label="eq:t_entropy" align="center" raw="h\left( X \right) = \frac{\nu +1}{2} \left[\psi\left({\frac{1+\nu }{2}}\right)-\psi\left({\frac{\nu }{2}}\right)\right]+\ln{\left[{\sqrt {\nu }}B\left({\frac{\nu }{2}},{\frac{1}{2}}\right)\right]}" alt="Differential entropy for a Student's t distribution."> -->

<div class="equation" align="center" data-raw-text="h\left( X \right) = \frac{\nu +1}{2} \left[\psi\left({\frac{1+\nu }{2}}\right)-\psi\left({\frac{\nu }{2}}\right)\right]+\ln{\left[{\sqrt {\nu }}B\left({\frac{\nu }{2}},{\frac{1}{2}}\right)\right]}" data-equation="eq:t_entropy">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/t/entropy/docs/img/equation_t_entropy.svg" alt="Differential entropy for a Student's t distribution.">
    <br>
</div>

<!-- </equation> -->

where `Β` and `ψ` denote the [beta][beta-function] and [digamma][digamma] functions, respectively.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### entropy( v )

Returns the [differential entropy][entropy] of a [Student's t][t-distribution] distribution with degrees of freedom `v` (in [nats][nats]).

```javascript
var y = entropy( 9.0 );
// returns ~1.533

y = entropy( 3.5 );
// returns ~1.721
```

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

```javascript
var y = entropy( -1.0 );
// returns NaN

y = entropy( 0.0 );
// returns NaN
```

</section>

<!-- /.usage -->

<!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- Package usage examples. -->

<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 entropy = require( '@stdlib/stats-base-dists-t-entropy' );

var opts = {
    'dtype': 'float64'
};
var v = uniform( 10, 0.0, 20.0, opts );

logEachMap( 'v: %0.4f, h(X;v): %0.4f', v, entropy );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

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<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/t/entropy.h"
```

#### stdlib_base_dists_t_entropy( v )

Returns the differential entropy of a Student's t distribution.

```c
double out = stdlib_base_dists_t_entropy( 9.0 );
// returns ~1.553
```

The function accepts the following arguments:

-   **v**: `[in] double` degrees of freedom.

```c
double stdlib_base_dists_t_entropy( const double v );
```

</section>

<!-- /.usage -->

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<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/t/entropy.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v*(max-min) );
}

int main( void ) {
    double v;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        v = random_uniform( 0.0, 20.0 );
        y = stdlib_base_dists_t_entropy( v );
        printf( "v: %lf, h(X;v): %lf\n", v, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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</section>

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<section class="related">

</section>

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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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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-base-dists-t-entropy/main/LICENSE

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

[entropy]: https://en.wikipedia.org/wiki/Entropy_%28information_theory%29

[nats]: https://en.wikipedia.org/wiki/Nat_%28unit%29

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

[digamma]: https://en.wikipedia.org/wiki/Digamma_function

</section>

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