@stdlib/stats/base/dists/beta/ctor/README.md

Standard library statistical functions.

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# Beta

> Beta distribution constructor.

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## Usage

```javascript
var Beta = require( '@stdlib/stats/base/dists/beta/ctor' );
```

#### Beta( \[alpha, beta] )

Returns a [beta][beta-distribution] distribution object.

```javascript
var beta = new Beta();

var mu = beta.mean;
// returns 0.5
```

By default, `alpha = 1.0` and `beta = 1.0`. To create a distribution having a different `alpha` (first shape parameter) and `beta` (second shape parameter), provide the corresponding arguments.

```javascript
var beta = new Beta( 2.0, 4.0 );

var mu = beta.mean;
// returns ~0.333
```

* * *

## beta

A [beta][beta-distribution] distribution object has the following properties and methods...

### Writable Properties

#### beta.alpha

First shape parameter of the distribution. `alpha` **must** be a positive number.

```javascript
var beta = new Beta();

var alpha = beta.alpha;
// returns 1.0

beta.alpha = 3.0;

alpha = beta.alpha;
// returns 3.0
```

#### beta.beta

Second shape parameter of the distribution. `beta` **must** be a positive number.

```javascript
var beta = new Beta( 2.0, 4.0 );

var b = beta.beta;
// returns 4.0

beta.beta = 3.0;

b = beta.beta;
// returns 3.0
```

* * *

### Computed Properties

#### Beta.prototype.entropy

Returns the [differential entropy][entropy].

```javascript
var beta = new Beta( 4.0, 12.0 );

var entropy = beta.entropy;
// returns ~-0.869
```

#### Beta.prototype.kurtosis

Returns the [excess kurtosis][kurtosis].

```javascript
var beta = new Beta( 4.0, 12.0 );

var kurtosis = beta.kurtosis;
// returns ~0.082
```

#### Beta.prototype.mean

Returns the [expected value][expected-value].

```javascript
var beta = new Beta( 4.0, 12.0 );

var mu = beta.mean;
// returns 0.25
```

#### Beta.prototype.median

Returns the [median][median].

```javascript
var beta = new Beta( 4.0, 12.0 );

var median = beta.median;
// returns ~0.239
```

#### Beta.prototype.mode

Returns the [mode][mode].

```javascript
var beta = new Beta( 4.0, 12.0 );

var mode = beta.mode;
// returns ~0.214
```

#### Beta.prototype.skewness

Returns the [skewness][skewness].

```javascript
var beta = new Beta( 4.0, 12.0 );

var skewness = beta.skewness;
// returns ~0.529
```

#### Beta.prototype.stdev

Returns the [standard deviation][standard-deviation].

```javascript
var beta = new Beta( 4.0, 12.0 );

var s = beta.stdev;
// returns ~0.105
```

#### Beta.prototype.variance

Returns the [variance][variance].

```javascript
var beta = new Beta( 4.0, 12.0 );

var s2 = beta.variance;
// returns ~0.011
```

* * *

### Methods

#### Beta.prototype.cdf( x )

Evaluates the [cumulative distribution function][cdf] (CDF).

```javascript
var beta = new Beta( 2.0, 4.0 );

var y = beta.cdf( 0.5 );
// returns ~0.813
```

#### Beta.prototype.logcdf( x )

Evaluates the natural logarithm of the [cumulative distribution function][cdf] (CDF).

```javascript
var beta = new Beta( 2.0, 4.0 );

var y = beta.logcdf( 0.5 );
// returns ~-0.208
```

#### Beta.prototype.logpdf( x )

Evaluates the natural logarithm of the [probability density function][pdf] (PDF).

```javascript
var beta = new Beta( 2.0, 4.0 );

var y = beta.logpdf( 0.8 );
// returns ~-2.0557
```

#### Beta.prototype.mgf( t )

Evaluates the [moment-generating function][mgf] (MGF).

```javascript
var beta = new Beta( 2.0, 4.0 );

var y = beta.mgf( 0.5 );
// returns ~1.186
```

#### Beta.prototype.pdf( x )

Evaluates the [probability density function][pdf] (PDF).

```javascript
var beta = new Beta( 2.0, 4.0 );

var y = beta.pdf( 0.8 );
// returns ~0.128
```

#### Beta.prototype.quantile( p )

Evaluates the [quantile function][quantile-function] at probability `p`.

```javascript
var beta = new Beta( 2.0, 4.0 );

var y = beta.quantile( 0.5 );
// returns ~0.314

y = beta.quantile( 1.9 );
// returns NaN
```

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* * *

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## Examples

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```javascript
var Beta = require( '@stdlib/stats/base/dists/beta/ctor' );

var beta = new Beta( 2.0, 4.0 );

var mu = beta.mean;
// returns ~0.333

var median = beta.median;
// returns ~0.314

var s2 = beta.variance;
// returns ~0.032

var y = beta.cdf( 0.8 );
// returns ~0.993
```

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[beta-distribution]: https://en.wikipedia.org/wiki/Beta_distribution

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

[mgf]: https://en.wikipedia.org/wiki/Moment-generating_function

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

[quantile-function]: https://en.wikipedia.org/wiki/Quantile_function

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

[expected-value]: https://en.wikipedia.org/wiki/Expected_value

[kurtosis]: https://en.wikipedia.org/wiki/Kurtosis

[median]: https://en.wikipedia.org/wiki/Median

[mode]: https://en.wikipedia.org/wiki/Mode_%28statistics%29

[skewness]: https://en.wikipedia.org/wiki/Skewness

[standard-deviation]: https://en.wikipedia.org/wiki/Standard_deviation

[variance]: https://en.wikipedia.org/wiki/Variance

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