README.md (3488B)
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 > [Gumbel][gumbel-distribution] distribution [probability density function (PDF)][pdf]. 24 25 <section class="intro"> 26 27 The [probability density function][pdf] (PDF) for a [Gumbel][gumbel-distribution] random variable is 28 29 <!-- <equation class="equation" label="eq:gumbel_pdf" align="center" raw="f(x;\mu,\beta) = \frac{1}{\beta}e^{-\left( \frac{x-\mu}{\beta}+e^{- \frac{x-\mu}{\beta}}\right)}" alt="Probability density function (PDF) for a Gumbel distribution."> --> 30 31 <div class="equation" align="center" data-raw-text="f(x;\mu,\beta) = \frac{1}{\beta}e^{-\left( \frac{x-\mu}{\beta}+e^{- \frac{x-\mu}{\beta}}\right)}" data-equation="eq:gumbel_pdf"> 32 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/gumbel/pdf/docs/img/equation_gumbel_pdf.svg" alt="Probability density function (PDF) for a Gumbel distribution."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 where `mu` is the location parameter and `beta > 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/gumbel/pdf' ); 50 ``` 51 52 #### pdf( x, mu, beta ) 53 54 Evaluates the [probability density function][pdf] (PDF) for a [Gumbel][gumbel-distribution] distribution with parameters `mu` (location parameter) and `beta` (scale parameter). 55 56 ```javascript 57 var y = pdf( 0.0, 0.0, 2.0 ); 58 // returns ~0.184 59 60 y = pdf( 0.0, 0.0, 1.0 ); 61 // returns ~0.368 62 63 y = pdf( 1.0, 3.0, 2.0 ); 64 // returns ~0.09 65 ``` 66 67 If provided `NaN` as any argument, the function returns `NaN`. 68 69 ```javascript 70 var y = pdf( NaN, 0.0, 1.0 ); 71 // returns NaN 72 73 y = pdf( 0.0, NaN, 1.0 ); 74 // returns NaN 75 76 y = pdf( 0.0, 0.0, NaN ); 77 // returns NaN 78 ``` 79 80 If provided `beta <= 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, 8.0, 0.0 ); 87 // returns NaN 88 ``` 89 90 #### pdf.factory( mu, beta ) 91 92 Partially apply `mu` and `beta` to create a reusable `function` for evaluating the PDF. 93 94 ```javascript 95 var mypdf = pdf.factory( 10.0, 2.0 ); 96 97 y = mypdf( 10.0 ); 98 // returns ~0.184 99 100 y = mypdf( 12.0 ); 101 // returns ~0.127 102 ``` 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 pdf = require( '@stdlib/stats/base/dists/gumbel/pdf' ); 117 118 var beta; 119 var mu; 120 var x; 121 var y; 122 var i; 123 124 for ( i = 0; i < 10; i++ ) { 125 x = randu() * 10.0; 126 mu = randu() * 10.0; 127 beta = randu() * 10.0; 128 y = pdf( x, mu, beta ); 129 console.log( 'x: %d, µ: %d, β: %d, f(x;µ,β): %d', x, mu, beta, y ); 130 } 131 ``` 132 133 </section> 134 135 <!-- /.examples --> 136 137 <section class="links"> 138 139 [gumbel-distribution]: https://en.wikipedia.org/wiki/Gumbel_distribution 140 141 [pdf]: https://en.wikipedia.org/wiki/Probability_density_function 142 143 </section> 144 145 <!-- /.links -->