time-to-botec

Benchmark sampling in different programming languages
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README.md (3693B)

      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
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     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 # Probability Density Function
     22 
     23 > [Cauchy][cauchy-distribution] distribution probability density function (PDF).
     24 
     25 <section class="intro">
     26 
     27 The [probability density function][pdf] (PDF) for a [Cauchy][cauchy-distribution] random variable is
     28 
     29 <!-- <equation class="equation" label="eq:cauchy_cauchy_pdf" align="center" raw="f(x;\gamma,x_0)=\frac{1}{\pi\gamma\,\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}\!" alt="Probability density function (PDF) for a Cauchy distribution."> -->
     30 
     31 <div class="equation" align="center" data-raw-text="f(x;\gamma,x_0)=\frac{1}{\pi\gamma\,\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}\!" data-equation="eq:cauchy_cauchy_pdf">
     32     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/cauchy/pdf/docs/img/equation_cauchy_cauchy_pdf.svg" alt="Probability density function (PDF) for a Cauchy distribution.">
     33     <br>
     34 </div>
     35 
     36 <!-- </equation> -->
     37 
     38 where `x0` is the location parameter and `gamma > 0` is the scale parameter.
     39 
     40 </section>
     41 
     42 <!-- /.intro -->
     43 
     44 <section class="usage">
     45 
     46 ## Usage
     47 
     48 ```javascript
     49 var pdf = require( '@stdlib/stats/base/dists/cauchy/pdf' );
     50 ```
     51 
     52 #### pdf( x, x0, gamma )
     53 
     54 Evaluates the [probability density function][pdf] (PDF) for a [Cauchy][cauchy-distribution] distribution with parameters `x0` (location parameter) and `gamma > 0` (scale parameter).
     55 
     56 ```javascript
     57 var y = pdf( 2.0, 1.0, 1.0 );
     58 // returns ~0.159
     59 
     60 y = pdf( 4.0, 3.0, 0.1 );
     61 // returns ~0.0315
     62 
     63 y = pdf( 4.0, 3.0, 3.0 );
     64 // returns ~0.095
     65 ```
     66 
     67 If provided `NaN` as any argument, the function returns `NaN`.
     68 
     69 ```javascript
     70 var y = pdf( NaN, 1.0, 1.0 );
     71 // returns NaN
     72 
     73 y = pdf( 2.0, NaN, 1.0 );
     74 // returns NaN
     75 
     76 y = pdf( 2.0, 1.0, NaN );
     77 // returns NaN
     78 ```
     79 
     80 If provided `gamma <= 0`, the function returns `NaN`.
     81 
     82 ```javascript
     83 var y = pdf( 2.0, 0.0, -1.0 );
     84 // returns NaN
     85 
     86 y = pdf( 2.0, 0.0, 0.0 );
     87 // returns NaN
     88 ```
     89 
     90 #### pdf.factory( x0, gamma )
     91 
     92 Returns a `function` for evaluating the [PDF][pdf] of a [Cauchy][cauchy-distribution] distribution with location parameter `x0` and scale parameter `gamma`.
     93 
     94 ```javascript
     95 var mypdf = pdf.factory( 10.0, 2.0 );
     96 
     97 var y = mypdf( 10.0 );
     98 // returns ~0.159
     99 
    100 y = mypdf( 5.0 );
    101 // returns ~0.022
    102 ```
    103 
    104 </section>
    105 
    106 <!-- /.usage -->
    107 
    108 <section class="examples">
    109 
    110 ## Examples
    111 
    112 <!-- eslint no-undef: "error" -->
    113 
    114 ```javascript
    115 var randu = require( '@stdlib/random/base/randu' );
    116 var EPS = require( '@stdlib/constants/float64/eps' );
    117 var pdf = require( '@stdlib/stats/base/dists/cauchy/pdf' );
    118 
    119 var gamma;
    120 var x0;
    121 var x;
    122 var y;
    123 var i;
    124 
    125 for ( i = 0; i < 10; i++ ) {
    126     x = randu() * 10.0;
    127     x0 = ( randu()*10.0 ) - 5.0;
    128     gamma = ( randu()*20.0 ) + EPS;
    129     y = pdf( x, gamma, x0 );
    130     console.log( 'x: %d, x0: %d, γ: %d, f(x;x0,γ): %d', x.toFixed(4), x0.toFixed(4), gamma.toFixed(4), y.toFixed(4) );
    131 }
    132 ```
    133 
    134 </section>
    135 
    136 <!-- /.examples -->
    137 
    138 <section class="links">
    139 
    140 [pdf]: https://en.wikipedia.org/wiki/Probability_density_function
    141 
    142 [cauchy-distribution]: https://en.wikipedia.org/wiki/Cauchy_distribution
    143 
    144 </section>
    145 
    146 <!-- /.links -->