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

Standard library statistical functions.

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# Moment-Generating Function

> [Logistic][logistic-distribution] distribution moment-generating function (MGF).

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The [moment-generating function][mgf] for a [logistic][logistic-distribution] random variable is

<!-- <equation class="equation" label="eq:logistic_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = e^{\mu t}\operatorname{B}(1-st, 1+st)" alt="Moment-generating function (MGF) for a logistic distribution."> -->

<div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = e^{\mu t}\operatorname{B}(1-st, 1+st)" data-equation="eq:logistic_mgf_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/logistic/mgf/docs/img/equation_logistic_mgf_function.svg" alt="Moment-generating function (MGF) for a logistic distribution.">
    <br>
</div>

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for `st ∈ (-1,1)`, where `mu` is the location parameter and `s` is the scale parameter. In above equation, `B` denotes the [Beta function][@stdlib/math/base/special/beta]. For `st` outside the interval `(-1,1)`, the function is not defined.

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## Usage

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

#### mgf( t, mu, s )

Evaluates the logarithm of the [moment-generating function][mgf] (MGF) for a [logistic][logistic-distribution] distribution with parameters `mu` (location parameter) and `s` (scale parameter).

```javascript
var y = mgf( 0.9, 0.0, 1.0 );
// returns ~9.15

y = mgf( 0.1, 4.0, 4.0 );
// returns ~1.971

y = mgf( -0.2, 4.0, 4.0 );
// returns ~1.921
```

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 provided `s < 0` or `abs( s * t ) > 1`, the function returns `NaN`.

```javascript
var y = mgf( 0.5, 0.0, -1.0 );
// returns NaN

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

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

Returns a function for evaluating the [moment-generating function][mgf] (MGF) of a [logistic][logistic-distribution] distribution with parameters `mu` (location parameter) and `s` (scale parameter).

```javascript
var mymgf = mgf.factory( 10.0, 0.5 );

var y = mymgf( 0.5 );
// returns ~164.846

y = mymgf( 2.0 );
// returns Infinity
```

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## Examples

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

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

var mu;
var s;
var t;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    t = randu();
    mu = (randu() * 10.0) - 5.0;
    s = randu() * 2.0;
    y = mgf( t, mu, s );
    console.log( 't: %d, µ: %d, s: %d, M_X(t;µ,s): %d', t.toFixed( 4 ), mu.toFixed( 4 ), s.toFixed( 4 ), y.toFixed( 4 ) );
}
```

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[@stdlib/math/base/special/beta]: https://www.npmjs.com/package/@stdlib/math-base-special-beta

[logistic-distribution]: https://en.wikipedia.org/wiki/Logistic_distribution

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

</section>

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