time-to-botec

README.md (5712B)

---

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 @license Apache-2.0
 
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 # Moment-Generating Function
 
 > [Negative binomial][negative-binomial-distribution] distribution moment-generating function (MGF).
 
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 <section class="intro">
 
 The [moment-generating function][mgf] for a [negative binomial][negative-binomial-distribution] random variable is
 
 <!-- <equation class="equation" label="eq:negative_binomial_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{\left( 1- p \right) e^t }{1 - p e^t}\biggr)^{\!r} \text{ for }t<-\log p" alt="Moment-generating function (MGF) for a negative binomial distribution."> -->
 
 <div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{\left( 1- p \right) e^t }{1 - p e^t}\biggr)^{\!r} \text{ for }t&lt;-\log p" data-equation="eq:negative_binomial_mgf_function">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/docs/img/equation_negative_binomial_mgf_function.svg" alt="Moment-generating function (MGF) for a negative binomial distribution.">
     <br>
 </div>
 
 <!-- </equation> -->
 
 where `r > 0` is the number of failures until the experiment is stopped and `0 <= p <= 1` is the success probability.
 
 </section>
 
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 <!-- Package usage documentation. -->
 
 <section class="usage">
 
 ## Usage
 
 ```javascript
 var mgf = require( '@stdlib/stats/base/dists/negative-binomial/mgf' );
 ```
 
 #### mgf( t, r, p )
 
 Evaluates the [moment-generating function][mgf] for a [negative binomial][negative-binomial-distribution] distribution with number of successes until experiment is stopped `r` and success probability `p`.
 
 ```javascript
 var y = mgf( 0.05, 20.0, 0.8 );
 // returns ~267.839
 
 y = mgf( 0.1, 20.0, 0.1 );
 // returns ~9.347
 ```
 
 While `r` can be interpreted as the number of successes until the experiment is stopped, the [negative binomial][negative-binomial-distribution] distribution is also defined for non-integers `r`. In this case, `r` denotes shape parameter of the [gamma mixing distribution][negative-binomial-mixture-representation].
 
 ```javascript
 var y = mgf( 0.1, 15.5, 0.5 );
 // returns ~26.375
 
 y = mgf( 0.5, 7.4, 0.4 );
 // returns ~2675.677
 ```
 
 If `t >= -ln( p )`, the function returns `NaN`.
 
 ```javascript
 var y = mgf( 0.7, 15.5, 0.5 ); // -ln( p ) = ~0.693
 // returns NaN
 ```
 
 If provided a `r` which is not a positive number, the function returns `NaN`.
 
 ```javascript
 var y = mgf( 0.2, 0.0, 0.5 );
 // returns NaN
 
 y = mgf( 0.2, -2.0, 0.5 );
 // returns NaN
 ```
 
 If provided `NaN` as any argument, the function returns `NaN`.
 
 ```javascript
 var y = mgf( NaN, 20.0, 0.5 );
 // returns NaN
 
 y = mgf( 0.0, NaN, 0.5 );
 // returns NaN
 
 y = mgf( 0.0, 20.0, NaN );
 // returns NaN
 ```
 
 If provided a success probability `p` outside of `[0,1]`, the function returns `NaN`.
 
 ```javascript
 var y = mgf( 0.2, 20, -1.0 );
 // returns NaN
 
 y = mgf( 0.2, 20, 1.5 );
 // returns NaN
 ```
 
 #### mgf.factory( r, p )
 
 Returns a function for evaluating the [moment-generating function][mgf] of  a [negative binomial][negative-binomial-distribution] distribution with number of successes until experiment is stopped `r` and success probability `p`.
 
 ```javascript
 var myMGF = mgf.factory( 4.3, 0.4 );
 var y = myMGF( 0.2 );
 // returns ~4.696
 
 y = myMGF( 0.4 );
 // returns ~30.83
 ```
 
 </section>
 
 <!-- /.usage -->
 
 <!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
 
 <section class="notes">
 
 </section>
 
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 <!-- Package usage examples. -->
 
 <section class="examples">
 
 ## 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/negative-binomial/mgf' );
 
 var p;
 var r;
 var t;
 var y;
 var i;
 
 for ( i = 0; i < 10; i++ ) {
     t = (randu() * 1.0) - 0.5;
     r = randu() * 50;
     p = randu();
     y = mgf( t, r, p );
     console.log( 't: %d, r: %d, p: %d, M_X(t;r,p): %d', t, r.toFixed( 4 ), p.toFixed( 4 ), y.toFixed( 4 ) );
 }
 ```
 
 </section>
 
 <!-- /.examples -->
 
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 <section class="references">
 
 </section>
 
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 <section class="links">
 
 [mgf]: https://en.wikipedia.org/wiki/Moment-generating_function
 
 [negative-binomial-mixture-representation]: https://en.wikipedia.org/wiki/Negative_binomial_distribution#Gamma.E2.80.93Poisson_mixture
 
 [negative-binomial-distribution]: https://en.wikipedia.org/wiki/Negative_binomial_distribution
 
 </section>
 
 <!-- /.links -->