README.md (4145B)
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 # Probability Density Function 22 23 > [Fréchet][frechet-distribution] distribution [probability density function][pdf]. 24 25 <section class="intro"> 26 27 The [probability density function][pdf] for a [Fréchet][frechet-distribution] random variable is 28 29 <!-- <equation class="equation" label="eq:frechet_pdf" align="center" raw="f\left( x; \mu, \beta \right ) = {\frac{\alpha }{s}}\;\left({\frac{x-m}{s}}\right)^{{-1-\alpha }}\;e^{{-({\frac{x-m}{s}})^{-\alpha}}}" alt="Probability density function for a Fréchet distribution."> --> 30 31 <div class="equation" align="center" data-raw-text="f\left( x; \mu, \beta \right ) = {\frac{\alpha }{s}}\;\left({\frac{x-m}{s}}\right)^{{-1-\alpha }}\;e^{{-({\frac{x-m}{s}})^{-\alpha}}}" data-equation="eq:frechet_pdf"> 32 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/frechet/pdf/docs/img/equation_frechet_pdf.svg" alt="Probability density function for a Fréchet distribution."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 where `alpha > 0` is the shape, `s > 0` the scale and `m` the location 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/frechet/pdf' ); 50 ``` 51 52 #### pdf( x, alpha, s, m ) 53 54 Evaluates the [probability density function][pdf] (PDF) for a [Fréchet][frechet-distribution] distribution with shape `alpha`, scale `s`, and location `m` at a value `x`. 55 56 ```javascript 57 var y = pdf( 10.0, 2.0, 3.0, 5.0 ); 58 // returns ~0.1 59 60 y = pdf( -3.0, 1.0, 2.0, -4.0 ); 61 // returns ~0.271 62 63 y = pdf( 0.0, 2.0, 1.0, -1.0 ); 64 // returns ~0.736 65 ``` 66 67 If provided `x <= m`, the function returns `0`. 68 69 ```javascript 70 y = pdf( -2.0, 2.0, 1.0, -1.0 ); 71 // returns 0.0 72 ``` 73 74 If provided `NaN` as any argument, the function returns `NaN`. 75 76 ```javascript 77 var y = pdf( NaN, 1.0, 1.0, 0.0 ); 78 // returns NaN 79 80 y = pdf( 0.0, NaN, 1.0, 0.0 ); 81 // returns NaN 82 83 y = pdf( 0.0, 1.0, NaN, 0.0); 84 // returns NaN 85 86 y = pdf( 0.0, 1.0, 1.0, NaN ); 87 // returns NaN 88 ``` 89 90 If provided `alpha <= 0`, the function returns `NaN`. 91 92 ```javascript 93 var y = pdf( 2.0, -0.1, 1.0, 1.0 ); 94 // returns NaN 95 96 y = pdf( 2.0, 0.0, 1.0, 1.0 ); 97 // returns NaN 98 ``` 99 100 If provided `s <= 0`, the function returns `NaN`. 101 102 ```javascript 103 var y = pdf( 2.0, 1.0, -1.0, 1.0 ); 104 // returns NaN 105 106 y = pdf( 2.0, 1.0, 0.0, 1.0 ); 107 // returns NaN 108 ``` 109 110 #### pdf.factory( alpha, s, m ) 111 112 Returns a function for evaluating the [probability density function][pdf] of a [Fréchet][frechet-distribution] distribution with shape `alpha`, scale `s`, and location `m`. 113 114 ```javascript 115 var mypdf = pdf.factory( 3.0, 3.0, 5.0 ); 116 117 var y = mypdf( 10.0 ); 118 // returns ~0.104 119 120 y = mypdf( 7.0 ); 121 // returns ~0.173 122 ``` 123 124 </section> 125 126 <!-- /.usage --> 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 pdf = require( '@stdlib/stats/base/dists/frechet/pdf' ); 137 138 var alpha; 139 var m; 140 var s; 141 var x; 142 var y; 143 var i; 144 145 for ( i = 0; i < 100; i++ ) { 146 alpha = randu() * 10.0; 147 x = randu() * 10.0; 148 s = randu() * 10.0; 149 m = randu() * 10.0; 150 y = pdf( x, alpha, s, m ); 151 console.log( 'x: %d, α: %d, s: %d, m: %d, f(x;α,s,m): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), s.toFixed( 4 ), m.toFixed( 4 ), y.toFixed( 4 ) ); 152 } 153 ``` 154 155 </section> 156 157 <!-- /.examples --> 158 159 <section class="links"> 160 161 [frechet-distribution]: https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution 162 163 [pdf]: https://en.wikipedia.org/wiki/Probability_density_function 164 165 </section> 166 167 <!-- /.links -->