@stdlib/stats/base/dists/erlang/mgf/README.md

Standard library statistical functions.

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# Moment-Generating Function

> [Erlang][erlang-distribution] distribution moment-generating function (MGF).

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The [moment-generating function][mgf] for an [Erlang][erlang-distribution] random variable is

<!-- <equation class="equation" label="eq:erlang_mgf" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \left(1 \,-\, \frac{t}{\lambda}\right)^{-k}" alt="Moment-generating function (MGF) for an Erlang distribution."> -->

<div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \left(1 \,-\, \frac{t}{\lambda}\right)^{-k}" data-equation="eq:erlang_mgf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/erlang/mgf/docs/img/equation_erlang_mgf.svg" alt="Moment-generating function (MGF) for an Erlang distribution.">
    <br>
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for `t < lambda`, where the nonnegative integer `k` is the shape parameter and `lambda > 0` is the rate parameter of the distribution. In the case that `t >= lambda`, the MGF is not defined.

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## Usage

```javascript
var mgf = require( '@stdlib/stats/base/dists/erlang/mgf' );
```

#### mgf( t, k, lambda )

Evaluates the [moment-generating function][mgf] (mgf) for an [Erlang][erlang-distribution] distribution with parameters `k` (shape parameter) and `lambda` (rate parameter).

```javascript
var y = mgf( 0.3, 1, 1.0 );
// returns ~1.429

y = mgf( 2.0, 2, 3.0 );
// returns ~9.0

y = mgf( -1.0, 2, 2.0 );
// returns ~0.444
```

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

```javascript
var y = mgf( NaN, 1, 1.0 );
// returns NaN

y = mgf( 0.0, NaN, 1.0 );
// returns NaN

y = mgf( 0.0, 1, NaN );
// returns NaN
```

If not provided a nonnegative integer for `k`, the function returns `NaN`.

```javascript
var y = mgf( 0.2, -2, 0.5 );
// returns NaN

y = mgf( 0.2, 0.5, 0.5 );
// returns NaN
```

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

```javascript
var y = mgf( 0.2, 1, 0.0 );
// returns NaN

y = mgf( 0.2, 1, -5.0 );
// returns NaN
```

#### mgf.factory( k, lambda )

Returns a function for evaluating the [moment-generating function][mgf] for an [Erlang][erlang-distribution] distribution with parameters `k` (shape parameter) and `lambda` (rate parameter).

```javascript
var myMGF = mgf.factory( 2, 0.5 );

var y = myMGF( 0.2 );
// returns ~2.778

y = myMGF( -0.5 );
// returns 0.25
```

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</section>

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## Examples

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

```javascript
var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var mgf = require( '@stdlib/stats/base/dists/erlang/mgf' );

var lambda;
var k;
var t;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    k = round( randu() * 10.0 );
    lambda = randu() * 10.0;
    t = randu() * lambda;
    y = mgf( t, k, lambda );
    console.log( 't: %d, k: %d, λ: %d, M_X(t;k,λ): %d', t.toFixed( 4 ), k, lambda.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</section>

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[erlang-distribution]: https://en.wikipedia.org/wiki/Erlang_distribution

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

</section>

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