README.md (4933B)
1 <!-- 2 3 @license Apache-2.0 4 5 Copyright (c) 2018 The Stdlib Authors. 6 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 10 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 # Variance 22 23 > [Kumaraswamy's double bounded][kumaraswamy-distribution] distribution [variance][variance]. 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 [variance][variance] for a [Kumaraswamy's double bounded][kumaraswamy-distribution] random variable with first shape parameter `a` and second shape parameter `b` is 30 31 <!-- <equation class="equation" label="eq:kumaraswamy_variance" align="center" raw="\operatorname{Var}\left( X \right) = m_2 - m_1^2" alt="Variance for a Kumaraswamy's double bounded distribution."> --> 32 33 <div class="equation" align="center" data-raw-text="\operatorname{Var}\left( X \right) = m_2 - m_1^2" data-equation="eq:kumaraswamy_variance"> 34 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/kumaraswamy/variance/docs/img/equation_kumaraswamy_variance.svg" alt="Variance for a Kumaraswamy's double bounded distribution."> 35 <br> 36 </div> 37 38 <!-- </equation> --> 39 40 where the raw moments of the distribution are given by 41 42 <!-- <equation class="equation" label="eq:kumaraswamy_raw_moments" align="center" raw="m_n = b \, B\left(1+\tfrac{n}{a}, b \right)" alt="Raw moments for a Kumaraswamy's double bounded distribution."> --> 43 44 <div class="equation" align="center" data-raw-text="m_n = b \, B\left(1+\tfrac{n}{a}, b \right)" data-equation="eq:kumaraswamy_raw_moments"> 45 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/kumaraswamy/variance/docs/img/equation_kumaraswamy_raw_moments.svg" alt="Raw moments for a Kumaraswamy's double bounded distribution."> 46 <br> 47 </div> 48 49 <!-- </equation> --> 50 51 with `B` denoting the [beta function][beta-function]. 52 53 </section> 54 55 <!-- /.intro --> 56 57 <!-- Package usage documentation. --> 58 59 <section class="usage"> 60 61 ## Usage 62 63 ```javascript 64 var variance = require( '@stdlib/stats/base/dists/kumaraswamy/variance' ); 65 ``` 66 67 #### variance( a, b ) 68 69 Returns the [variance][variance] of a [Kumaraswamy's double bounded][kumaraswamy-distribution] distribution with first shape parameter `a` and second shape parameter `b`. 70 71 ```javascript 72 var v = variance( 1.0, 1.0 ); 73 // returns ~0.083 74 75 v = variance( 4.0, 12.0 ); 76 // returns ~0.017 77 78 v = variance( 2.0, 8.0 ); 79 // returns ~0.021 80 ``` 81 82 If provided `NaN` as any argument, the function returns `NaN`. 83 84 ```javascript 85 var v = variance( NaN, 2.0 ); 86 // returns NaN 87 88 v = variance( 2.0, NaN ); 89 // returns NaN 90 ``` 91 92 If provided `a <= 0`, the function returns `NaN`. 93 94 ```javascript 95 var y = variance( -1.0, 0.5 ); 96 // returns NaN 97 98 y = variance( 0.0, 0.5 ); 99 // returns NaN 100 ``` 101 102 If provided `b <= 0`, the function returns `NaN`. 103 104 ```javascript 105 var y = variance( 0.5, -1.0 ); 106 // returns NaN 107 108 y = variance( 0.5, 0.0 ); 109 // returns NaN 110 ``` 111 112 </section> 113 114 <!-- /.usage --> 115 116 <!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. --> 117 118 <section class="notes"> 119 120 </section> 121 122 <!-- /.notes --> 123 124 <!-- Package usage examples. --> 125 126 <section class="examples"> 127 128 ## Examples 129 130 <!-- eslint no-undef: "error" --> 131 132 ```javascript 133 var randu = require( '@stdlib/random/base/randu' ); 134 var variance = require( '@stdlib/stats/base/dists/kumaraswamy/variance' ); 135 136 var a; 137 var b; 138 var v; 139 var i; 140 141 for ( i = 0; i < 10; i++ ) { 142 a = randu() * 10.0; 143 b = randu() * 10.0; 144 v = variance( a, b ); 145 console.log( 'a: %d, b: %d, Var(X;a,b): %d', a.toFixed( 4 ), b.toFixed( 4 ), v.toFixed( 4 ) ); 146 } 147 ``` 148 149 </section> 150 151 <!-- /.examples --> 152 153 <!-- 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. --> 154 155 <section class="references"> 156 157 </section> 158 159 <!-- /.references --> 160 161 <!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. --> 162 163 <section class="links"> 164 165 [beta-function]: https://en.wikipedia.org/wiki/Beta_function 166 167 [kumaraswamy-distribution]: https://en.wikipedia.org/wiki/Kumaraswamy_distribution 168 169 [variance]: https://en.wikipedia.org/wiki/Variance 170 171 </section> 172 173 <!-- /.links -->