time-to-botec

Benchmark sampling in different programming languages
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README.md (4352B)

      1 <!--
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      3 @license Apache-2.0
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      5 Copyright (c) 2018 The Stdlib Authors.
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      7 Licensed under the Apache License, Version 2.0 (the "License");
      8 you may not use this file except in compliance with the License.
      9 You may obtain a copy of the License at
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     11    http://www.apache.org/licenses/LICENSE-2.0
     12 
     13 Unless required by applicable law or agreed to in writing, software
     14 distributed under the License is distributed on an "AS IS" BASIS,
     15 WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
     16 See the License for the specific language governing permissions and
     17 limitations under the License.
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     19 -->
     20 
     21 # Moment-Generating Function
     22 
     23 > [Geometric][geometric-distribution] distribution moment-generating function (MGF).
     24 
     25 <!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->
     26 
     27 <section class="intro">
     28 
     29 The [moment-generating function][mgf] for a [geometric][geometric-distribution] random variable is
     30 
     31 <!-- <equation class="equation" label="eq:geometric_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \frac{pe^t}{1-(1-p) e^t} \text{ for } t<-\ln(1-p)" alt="Moment-generating function (MGF) for a geometric distribution."> -->
     32 
     33 <div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \frac{pe^t}{1-(1-p) e^t} \text{ for } t&lt;-\ln(1-p)" data-equation="eq:geometric_mgf_function">
     34     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/geometric/mgf/docs/img/equation_geometric_mgf_function.svg" alt="Moment-generating function (MGF) for a geometric distribution.">
     35     <br>
     36 </div>
     37 
     38 <!-- </equation> -->
     39 
     40 where `0 < p <= 1` is the success probability. For `t >= -ln(1-p)`, the MGF is not defined.
     41 
     42 </section>
     43 
     44 <!-- /.intro -->
     45 
     46 <!-- Package usage documentation. -->
     47 
     48 <section class="usage">
     49 
     50 ## Usage
     51 
     52 ```javascript
     53 var mgf = require( '@stdlib/stats/base/dists/geometric/mgf' );
     54 ```
     55 
     56 #### mgf( t, p )
     57 
     58 Evaluates the moment-generating function ([MGF][mgf]) of a [geometric][geometric-distribution] distribution with success probability `p`.
     59 
     60 ```javascript
     61 var y = mgf( 0.2, 0.5 );
     62 // returns ~1.569
     63 
     64 y = mgf( 0.4, 0.5 );
     65 // returns ~2.936
     66 ```
     67 
     68 If provided `NaN` as any argument, the function returns `NaN`.
     69 
     70 ```javascript
     71 var y = mgf( NaN, 0.0 );
     72 // returns NaN
     73 
     74 y = mgf( 0.0, NaN );
     75 // returns NaN
     76 ```
     77 
     78 If provided a success probability `p` outside of the interval `[0,1]`, the function returns `NaN`.
     79 
     80 ```javascript
     81 var y = mgf( -2.0, -1.0 );
     82 // returns NaN
     83 
     84 y = mgf( 0.2, 2.0 );
     85 // returns NaN
     86 ```
     87 
     88 If `t >= -ln(1-p)`, the function returns `NaN`.
     89 
     90 ```javascript
     91 var y = mgf( 0.8, 0.5 );
     92 // returns NaN
     93 ```
     94 
     95 #### mgf.factory( p )
     96 
     97 Returns a function for evaluating the [moment-generating function][mgf] of a [geometric][geometric-distribution] distribution with parameter `p` (success probability).
     98 
     99 ```javascript
    100 var mymgf = mgf.factory( 0.8 );
    101 var y = mymgf( -0.2 );
    102 // returns ~0.783
    103 ```
    104 
    105 </section>
    106 
    107 <!-- /.usage -->
    108 
    109 <!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
    110 
    111 <section class="notes">
    112 
    113 </section>
    114 
    115 <!-- /.notes -->
    116 
    117 <!-- Package usage examples. -->
    118 
    119 <section class="examples">
    120 
    121 ## Examples
    122 
    123 <!-- eslint no-undef: "error" -->
    124 
    125 ```javascript
    126 var randu = require( '@stdlib/random/base/randu' );
    127 var mgf = require( '@stdlib/stats/base/dists/geometric/mgf' );
    128 
    129 var p;
    130 var t;
    131 var y;
    132 var i;
    133 
    134 for ( i = 0; i < 10; i++ ) {
    135     t = randu();
    136     p = randu();
    137     y = mgf( t, p );
    138     console.log( 't: %d, p: %d, M_X(t;p): %d', t, p.toFixed( 4 ), y.toFixed( 4 ) );
    139 }
    140 ```
    141 
    142 </section>
    143 
    144 <!-- /.examples -->
    145 
    146 <!-- Section to include cited references. If references are included, add a horizontal rule *before* the section. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
    147 
    148 <section class="references">
    149 
    150 </section>
    151 
    152 <!-- /.references -->
    153 
    154 <!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
    155 
    156 <section class="links">
    157 
    158 [geometric-distribution]: https://en.wikipedia.org/wiki/Geometric_distribution
    159 
    160 [mgf]: https://en.wikipedia.org/wiki/Moment-generating_function
    161 
    162 </section>
    163 
    164 <!-- /.links -->