# Gamma

Gamma distribution constructor.

## Usage

``````var Gamma = require( '@stdlib/math/base/dists/gamma/ctor' );
``````

#### Gamma( [alpha, beta] )

Returns a gamma distribution object.

``````var gamma = new Gamma();

var mode = gamma.mode;
// returns 0.0
``````

By default, `alpha = 1.0` and `beta = 1.0`. To create a distribution having a different `alpha` (shape parameter) and `beta` (rate parameter), provide parameter values.

``````var gamma = new Gamma( 2.0, 4.0 );

var mu = gamma.mean;
// returns 1.0
``````

## gamma

A gamma distribution object has the following properties and methods...

### Writable Properties

#### gamma.alpha

Shape parameter of the distribution. `alpha` must be a positive number.

``````var gamma = new Gamma();

var alpha = gamma.alpha;
// returns 1.0

gamma.alpha = 3.0;

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

#### gamma.beta

Rate parameter of the distribution. `beta` must be a positive number.

``````var gamma = new Gamma( 2.0, 4.0 );

var b = gamma.beta;
// returns 4.0

gamma.beta = 3.0;

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

### Computed Properties

#### Gamma.prototype.entropy

Returns the differential entropy.

``````var gamma = new Gamma( 4.0, 12.0 );

var entropy = gamma.entropy;
// returns ~-0.462
``````

#### Gamma.prototype.kurtosis

Returns the excess kurtosis.

``````var gamma = new Gamma( 4.0, 12.0 );

var kurtosis = gamma.kurtosis;
// returns 1.5
``````

#### Gamma.prototype.mean

Returns the expected value.

``````var gamma = new Gamma( 4.0, 12.0 );

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

#### Gamma.prototype.mode

Returns the mode.

``````var gamma = new Gamma( 4.0, 12.0 );

var mode = gamma.mode;
// returns 0.25
``````

#### Gamma.prototype.skewness

Returns the skewness.

``````var gamma = new Gamma( 4.0, 12.0 );

var skewness = gamma.skewness;
// returns 1.0
``````

#### Gamma.prototype.stdev

Returns the standard deviation.

``````var gamma = new Gamma( 4.0, 12.0 );

var s = gamma.stdev;
// returns ~0.167
``````

#### Gamma.prototype.variance

Returns the variance.

``````var gamma = new Gamma( 4.0, 12.0 );

var s2 = gamma.variance;
// returns ~0.028
``````

### Methods

#### Gamma.prototype.cdf( x )

Evaluates the cumulative distribution function (CDF).

``````var gamma = new Gamma( 2.0, 4.0 );

var y = gamma.cdf( 0.5 );
// returns ~0.594
``````

#### Gamma.prototype.logpdf( x )

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

``````var gamma = new Gamma( 2.0, 4.0 );

var y = gamma.logpdf( 0.8 );
// returns ~-0.651
``````

#### Gamma.prototype.mgf( t )

Evaluates the moment-generating function (MGF).

``````var gamma = new Gamma( 2.0, 4.0 );

var y = gamma.mgf( 0.5 );
// returns ~1.306
``````

#### Gamma.prototype.pdf( x )

Evaluates the probability density function (PDF).

``````var gamma = new Gamma( 2.0, 4.0 );

var y = gamma.pdf( 0.8 );
// returns ~0.522
``````

#### Gamma.prototype.quantile( p )

Evaluates the quantile function at probability `p`.

``````var gamma = new Gamma( 2.0, 4.0 );

var y = gamma.quantile( 0.5 );
// returns ~0.42

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

## Examples

``````var Gamma = require( '@stdlib/math/base/dists/gamma/ctor' );

var gamma = new Gamma( 2.0, 4.0 );

var mu = gamma.mean;
// returns 0.5

var mode = gamma.mode;
// returns 0.25

var s2 = gamma.variance;
// returns 0.125

var y = gamma.cdf( 0.8 );
// returns ~0.829
``````