time-to-botec

Benchmark sampling in different programming languages
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README.md (4534B)

      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.
     18 
     19 -->
     20 
     21 # Moment-Generating Function
     22 
     23 > [Gumbel][gumbel-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 [Gumbel][gumbel-distribution] random variable is
     30 
     31 <!-- <equation class="equation" label="eq:gumbel_mgf" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \Gamma(1+\beta\,t)\, e^{\mu\,t}" alt="Moment-generating function (MGF) for a Gumbel distribution."> -->
     32 
     33 <div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \Gamma(1+\beta\,t)\, e^{\mu\,t}" data-equation="eq:gumbel_mgf">
     34     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/gumbel/mgf/docs/img/equation_gumbel_mgf.svg" alt="Moment-generating function (MGF) for a Gumbel distribution.">
     35     <br>
     36 </div>
     37 
     38 <!-- </equation> -->
     39 
     40 where `mu` is the location parameter and `beta > 0` is the scale parameter.
     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/gumbel/mgf' );
     54 ```
     55 
     56 #### mgf( t, mu, beta )
     57 
     58 Evaluates the [moment-generating function][mgf] (MGF) for a [Gumbel][gumbel-distribution] distribution with parameters `mu` (location parameter) and `beta > 0` (scale parameter).
     59 
     60 ```javascript
     61 var y = mgf( -1.0, 0.0, 3.0 );
     62 // returns 6.0
     63 
     64 y = mgf( 0.1, 0.0, 3.0 );
     65 // returns ~1.298
     66 
     67 y = mgf( 0.0, 0.0, 1.0 );
     68 // returns 1.0
     69 ```
     70 
     71 If provided `NaN` as any argument, the function returns `NaN`.
     72 
     73 ```javascript
     74 var y = mgf( NaN, 0.0, 1.0 );
     75 // returns NaN
     76 
     77 y = mgf( 0.0, NaN, 1.0 );
     78 // returns NaN
     79 
     80 y = mgf( 0.0, 0.0, NaN );
     81 // returns NaN
     82 ```
     83 
     84 If provided `t >= 1/beta`, the function returns `NaN`.
     85 
     86 ```javascript
     87 var y = mgf( 0.8, 0.0, 2.0 );
     88 // returns NaN
     89 ```
     90 
     91 If provided `beta <= 0`, the function returns `NaN`.
     92 
     93 ```javascript
     94 var y = mgf( 0.5, 0.0, -1.0 );
     95 // returns NaN
     96 
     97 y = mgf( 0.5, 0.0, 0.0 );
     98 // returns NaN
     99 ```
    100 
    101 #### mgf.factory( mu, beta )
    102 
    103 Returns a function for evaluating the [moment-generating function][mgf] of a [Gumbel][gumbel-distribution] distribution with parameters `mu` (location parameter) and `beta > 0` (scale parameter).
    104 
    105 ```javascript
    106 var myMGF = mgf.factory( 0.0, 2.0 );
    107 var y = myMGF( 0.2 );
    108 // returns ~1.489
    109 
    110 y = myMGF( -1.0 );
    111 // returns 2.0
    112 ```
    113 
    114 </section>
    115 
    116 <!-- /.usage -->
    117 
    118 <!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
    119 
    120 <section class="notes">
    121 
    122 </section>
    123 
    124 <!-- /.notes -->
    125 
    126 <!-- Package usage examples. -->
    127 
    128 <section class="examples">
    129 
    130 ## Examples
    131 
    132 <!-- eslint no-undef: "error" -->
    133 
    134 ```javascript
    135 var randu = require( '@stdlib/random/base/randu' );
    136 var mgf = require( '@stdlib/stats/base/dists/gumbel/mgf' );
    137 
    138 var beta;
    139 var mu;
    140 var t;
    141 var y;
    142 var i;
    143 
    144 for ( i = 0; i < 10; i++ ) {
    145     t = randu();
    146     mu = (randu() * 10.0) - 5.0;
    147     beta = randu() * 20.0;
    148     y = mgf( t, mu, beta );
    149     console.log( 't: %d, µ: %d, β: %d, M_X(t;µ,β): %d', t.toFixed( 4 ), mu.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
    150 }
    151 ```
    152 
    153 </section>
    154 
    155 <!-- /.examples -->
    156 
    157 <!-- 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. -->
    158 
    159 <section class="references">
    160 
    161 </section>
    162 
    163 <!-- /.references -->
    164 
    165 <!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
    166 
    167 <section class="links">
    168 
    169 [gumbel-distribution]: https://en.wikipedia.org/wiki/Gumbel_distribution
    170 
    171 [mgf]: https://en.wikipedia.org/wiki/Moment-generating_function
    172 
    173 </section>
    174 
    175 <!-- /.links -->