time-to-botec

Benchmark sampling in different programming languages
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README.md (3667B)

      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 # Cumulative Distribution Function
     22 
     23 > [Student's t][t-distribution] distribution [cumulative distribution function][cdf] (CDF).
     24 
     25 <section class="intro">
     26 
     27 The [cumulative distribution function][cdf] (CDF) for a [t distribution][t-distribution] random variable is
     28 
     29 <!-- <equation class="equation" label="eq:t_cdf" align="center" raw="F(x;\nu) = 1 - \frac{1}{2} \frac{\operatorname{Beta}(\tfrac{\nu}{\nu + x^2};\,\tfrac{\nu}{2},\tfrac{1}{2})}{\operatorname{Beta}(\tfrac{\nu}{2}, \tfrac{1}{2})}" alt="Cumulative distribution function (CDF) for a Student's t distribution."> -->
     30 
     31 <div class="equation" align="center" data-raw-text="F(x;\nu) = 1 - \frac{1}{2} \frac{\operatorname{Beta}(\tfrac{\nu}{\nu + x^2};\,\tfrac{\nu}{2},\tfrac{1}{2})}{\operatorname{Beta}(\tfrac{\nu}{2}, \tfrac{1}{2})}" data-equation="eq:t_cdf">
     32     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/t/cdf/docs/img/equation_t_cdf.svg" alt="Cumulative distribution function (CDF) for a Student's t distribution.">
     33     <br>
     34 </div>
     35 
     36 <!-- </equation> -->
     37 
     38 where `v > 0` is the degrees of freedom. 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/t/cdf' );
     50 ```
     51 
     52 #### cdf( x, v )
     53 
     54 Evaluates the [cumulative distribution function][cdf] (CDF) for a [Student's t][t-distribution] distribution with degrees of freedom `v`.
     55 
     56 ```javascript
     57 var y = cdf( 2.0, 0.1 );
     58 // returns ~0.611
     59 
     60 y = cdf( 1.0, 2.0 );
     61 // returns ~0.789
     62 
     63 y = cdf( -1.0, 4.0 );
     64 // returns ~0.187
     65 ```
     66 
     67 If provided `NaN` as any argument, the function returns `NaN`.
     68 
     69 ```javascript
     70 var y = cdf( NaN, 1.0 );
     71 // returns NaN
     72 
     73 y = cdf( 0.0, NaN );
     74 // returns NaN
     75 ```
     76 
     77 If provided `v <= 0`, the function returns `NaN`.
     78 
     79 ```javascript
     80 var y = cdf( 2.0, -1.0 );
     81 // returns NaN
     82 
     83 y = cdf( 2.0, 0.0 );
     84 // returns NaN
     85 ```
     86 
     87 #### cdf.factory( v )
     88 
     89 Returns a `function` for evaluating the [cdf][cdf] of a [Student's t][t-distribution] distribution with degrees of freedom `v`.
     90 
     91 ```javascript
     92 var mycdf = cdf.factory( 0.5 );
     93 var y = mycdf( 3.0 );
     94 // returns ~0.816
     95 
     96 y = mycdf( 1.0 );
     97 // returns ~0.699
     98 ```
     99 
    100 </section>
    101 
    102 <!-- /.usage -->
    103 
    104 <section class="examples">
    105 
    106 ## Examples
    107 
    108 <!-- eslint no-undef: "error" -->
    109 
    110 ```javascript
    111 var randu = require( '@stdlib/random/base/randu' );
    112 var cdf = require( '@stdlib/stats/base/dists/t/cdf' );
    113 
    114 var v;
    115 var x;
    116 var y;
    117 var i;
    118 
    119 for ( i = 0; i < 10; i++ ) {
    120     x = (randu() * 6.0) - 3.0;
    121     v = randu() * 10.0;
    122     y = cdf( x, v );
    123     console.log( 'x: %d, v: %d, F(x;v): %d', x.toFixed( 4 ), v.toFixed( 4 ), y.toFixed( 4 ) );
    124 }
    125 ```
    126 
    127 </section>
    128 
    129 <!-- /.examples -->
    130 
    131 <section class="links">
    132 
    133 [beta-function]: https://en.wikipedia.org/wiki/Beta_function
    134 
    135 [cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function
    136 
    137 [t-distribution]: https://en.wikipedia.org/wiki/Student%27s_t-distribution
    138 
    139 </section>
    140 
    141 <!-- /.links -->