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

Standard library statistical functions.

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# Moment-Generating Function

> [Laplace][laplace-distribution] distribution moment-generating function (MGF).

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The [moment-generating function][mgf] for a [Laplace (double exponential)][laplace-distribution] random variable is

<!-- <equation class="equation" label="eq:laplace_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \frac{\exp(\mu\,t)}{1-b^2\,t^2}\,\!\text{ for }|t|<1/b" alt="Moment-generating function (MGF) for a Laplace (double exponential) distribution."> -->

<div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \frac{\exp(\mu\,t)}{1-b^2\,t^2}\,\!\text{ for }|t|&lt;1/b" data-equation="eq:laplace_mgf_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/laplace/mgf/docs/img/equation_laplace_mgf_function.svg" alt="Moment-generating function (MGF) for a Laplace (double exponential) distribution.">
    <br>
</div>

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where `mu` is the location parameter and `b` is the scale parameter. For `|t| >= 1/b`, the [MGF][mgf] is undefined.

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## Usage

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

#### mgf( t, mu, b )

Evaluates the [moment-generating function][mgf] (MGF) for a [Laplace][laplace-distribution] (double exponential) distribution with parameters `mu` (location) and `b` (scale).

```javascript
var y = mgf( 0.5, 0.0, 1.0 );
// returns ~1.333

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

y = mgf( -1.0, 4.0, 0.2 );
// returns ~0.019
```

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

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

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

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

If `t` is not inside the interval `(-1/b, 1/b)`, the function returns `NaN`.

```javascript
var y = mgf( 1.0, 0.0, 2.0 );
// returns NaN

y = mgf( -0.5, 0.0, 4.0 );
// returns NaN
```

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

```javascript
var y = mgf( 2.0, 0.0, 0.0 );
// returns NaN

y = mgf( 2.0, 0.0, -1.0 );
// returns NaN
```

#### mgf.factory( mu, b )

Returns a function for evaluating the [moment-generating function][mgf] (MGF) of a [Laplace][laplace-distribution] (double exponential) distribution with parameters `mu` and `b`.

```javascript
var mymgf = mgf.factory( 4.0, 2.0 );

var y = mymgf( 0.2 );
// returns ~2.649

y = mymgf( 0.4 );
// returns ~13.758
```

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</section>

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## Examples

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

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

var mu;
var b;
var t;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    t = randu();
    mu = (randu() * 10.0) - 5.0;
    b = randu() * 20.0;
    y = mgf( t, mu, b );
    console.log( 't: %d, µ: %d, b: %d, M_X(t;µ,b): %d', t.toFixed( 4 ), mu.toFixed( 4 ), b.toFixed( 4 ), y.toFixed( 4 ) );
}
```

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[laplace-distribution]: https://en.wikipedia.org/wiki/Laplace_distribution

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

</section>

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