time-to-botec

Benchmark sampling in different programming languages
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README.md (4327B)

      1 <!--
      2 
      3 @license Apache-2.0
      4 
      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 # Cumulative Distribution Function
     22 
     23 > [Beta][beta-distribution] distribution [cumulative distribution function][cdf].
     24 
     25 <section class="intro">
     26 
     27 The [cumulative distribution function][cdf] for a [beta][beta-distribution] random variable is
     28 
     29 <!-- <equation class="equation" label="eq:beta_cdf" align="center" raw="F(x;\alpha,\beta) = \frac{\operatorname{Beta}(x;\alpha,\beta)}{\operatorname{Beta}(\alpha,\beta)}" alt="Cumulative distribution function for a beta distribution."> -->
     30 
     31 <div class="equation" align="center" data-raw-text="F(x;\alpha,\beta) = \frac{\operatorname{Beta}(x;\alpha,\beta)}{\operatorname{Beta}(\alpha,\beta)}" data-equation="eq:beta_cdf">
     32     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/beta/cdf/docs/img/equation_beta_cdf.svg" alt="Cumulative distribution function for a beta distribution.">
     33     <br>
     34 </div>
     35 
     36 <!-- </equation> -->
     37 
     38 where `alpha > 0` is the first shape parameter and `beta > 0` is the second shape parameter. In the definition, `Beta( x; a, b )` denotes the lower incomplete beta function and `Beta( a, b )` the [beta function][beta-function].
     39 
     40 </section>
     41 
     42 <!-- /.intro -->
     43 
     44 <section class="usage">
     45 
     46 ## Usage
     47 
     48 ```javascript
     49 var cdf = require( '@stdlib/stats/base/dists/beta/cdf' );
     50 ```
     51 
     52 #### cdf( x, alpha, beta )
     53 
     54 Evaluates the [cumulative distribution function][cdf] (CDF) for a [beta][beta-distribution] distribution with parameters `alpha` (first shape parameter) and `beta` (second shape parameter).
     55 
     56 ```javascript
     57 var y = cdf( 0.5, 1.0, 1.0 );
     58 // returns 0.5
     59 
     60 y = cdf( 0.5, 2.0, 4.0 );
     61 // returns ~0.813
     62 
     63 y = cdf( 0.2, 2.0, 2.0 );
     64 // returns ~0.104
     65 
     66 y = cdf( 0.8, 4.0, 4.0 );
     67 // returns ~0.967
     68 
     69 y = cdf( -0.5, 4.0, 2.0 );
     70 // returns 0.0
     71 
     72 y = cdf( -Infinity, 4.0, 2.0 );
     73 // returns 0.0
     74 
     75 y = cdf( 1.5, 4.0, 2.0 );
     76 // returns 1.0
     77 
     78 y = cdf( +Infinity, 4.0, 2.0 );
     79 // returns 1.0
     80 ```
     81 
     82 If provided `NaN` as any argument, the function returns `NaN`.
     83 
     84 ```javascript
     85 var y = cdf( NaN, 1.0, 1.0 );
     86 // returns NaN
     87 
     88 y = cdf( 0.0, NaN, 1.0 );
     89 // returns NaN
     90 
     91 y = cdf( 0.0, 1.0, NaN );
     92 // returns NaN
     93 ```
     94 
     95 If provided `alpha <= 0`, the function returns `NaN`.
     96 
     97 ```javascript
     98 var y = cdf( 2.0, -1.0, 0.5 );
     99 // returns NaN
    100 
    101 y = cdf( 2.0, 0.0, 0.5 );
    102 // returns NaN
    103 ```
    104 
    105 If provided `beta <= 0`, the function returns `NaN`.
    106 
    107 ```javascript
    108 var y = cdf( 2.0, 0.5, -1.0 );
    109 // returns NaN
    110 
    111 y = cdf( 2.0, 0.5, 0.0 );
    112 // returns NaN
    113 ```
    114 
    115 #### cdf.factory( alpha, beta )
    116 
    117 Returns a function for evaluating the [cumulative distribution function][cdf] for a [beta][beta-distribution] distribution with parameters `alpha` (first shape parameter) and `beta` (second shape parameter).
    118 
    119 ```javascript
    120 var mycdf = cdf.factory( 0.5, 0.5 );
    121 
    122 var y = mycdf( 0.8 );
    123 // returns ~0.705
    124 
    125 y = mycdf( 0.3 );
    126 // returns ~0.369
    127 ```
    128 
    129 </section>
    130 
    131 <!-- /.usage -->
    132 
    133 <section class="examples">
    134 
    135 ## Examples
    136 
    137 <!-- eslint no-undef: "error" -->
    138 
    139 ```javascript
    140 var randu = require( '@stdlib/random/base/randu' );
    141 var EPS = require( '@stdlib/constants/float64/eps' );
    142 var cdf = require( '@stdlib/stats/base/dists/beta/cdf' );
    143 
    144 var alpha;
    145 var beta;
    146 var x;
    147 var y;
    148 var i;
    149 
    150 for ( i = 0; i < 10; i++ ) {
    151     x = randu();
    152     alpha = ( randu()*5.0 ) + EPS;
    153     beta = ( randu()*5.0 ) + EPS;
    154     y = cdf( x, alpha, beta );
    155     console.log( 'x: %d, α: %d, β: %d, F(x;α,β): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
    156 }
    157 ```
    158 
    159 </section>
    160 
    161 <!-- /.examples -->
    162 
    163 <section class="links">
    164 
    165 [beta-distribution]: https://en.wikipedia.org/wiki/Beta_distribution
    166 
    167 [beta-function]: https://en.wikipedia.org/wiki/Beta_function
    168 
    169 [cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function
    170 
    171 </section>
    172 
    173 <!-- /.links -->