README.md (4116B)
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 > [Erlang][erlang-distribution] distribution probability density function (PDF). 24 25 <section class="intro"> 26 27 The [probability density function][pdf] (PDF) for an [Erlang][erlang-distribution] random variable is 28 29 <!-- <equation class="equation" label="eq:erlang_pdf" align="center" raw="f(x; k,\lambda)={\lambda^k x^{k-1} e^{-\lambda x} \over (k-1)!} 1(x \ge 0)" alt="Probability density function (PDF) for an Erlang distribution."> --> 30 31 <div class="equation" align="center" data-raw-text="f(x; k,\lambda)={\lambda^k x^{k-1} e^{-\lambda x} \over (k-1)!} 1(x \ge 0)" data-equation="eq:erlang_pdf"> 32 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/erlang/pdf/docs/img/equation_erlang_pdf.svg" alt="Probability density function (PDF) for an Erlang distribution."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 where `k` is the shape parameter and `lambda` is the rate 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/erlang/pdf' ); 50 ``` 51 52 #### pdf( x, k, lambda ) 53 54 Evaluates the [probability density function][pdf] (PDF) for an [Erlang][erlang-distribution] distribution with parameters `k` (shape parameter) and `lambda` (rate parameter). 55 56 ```javascript 57 var y = pdf( 0.1, 1, 1.0 ); 58 // returns ~0.905 59 60 y = pdf( 0.5, 2, 2.5 ); 61 // returns ~0.895 62 63 y = pdf( -1.0, 4, 2.0 ); 64 // returns 0.0 65 ``` 66 67 If provided `NaN` as any argument, the function returns `NaN`. 68 69 ```javascript 70 var y = pdf( NaN, 1, 1.0 ); 71 // returns NaN 72 73 y = pdf( 0.0, NaN, 1.0 ); 74 // returns NaN 75 76 y = pdf( 0.0, 1, NaN ); 77 // returns NaN 78 ``` 79 80 If not provided a nonnegative integer for `k`, the function returns `NaN`. 81 82 ```javascript 83 var y = pdf( 2.0, -2, 0.5 ); 84 // returns NaN 85 86 y = pdf( 2.0, 0.5, 0.5 ); 87 // returns NaN 88 ``` 89 90 If provided `k = 0`, the function evaluates the [PDF][pdf] of a [degenerate distribution][degenerate-distribution] centered at `0`. 91 92 ```javascript 93 var y = pdf( 2.0, 0.0, 2.0 ); 94 // returns 0.0 95 96 y = pdf( 0.0, 0.0, 2.0 ); 97 // returns Infinity 98 ``` 99 100 If provided `lambda <= 0`, the function returns `NaN`. 101 102 ```javascript 103 var y = pdf( 2.0, 1, 0.0 ); 104 // returns NaN 105 106 y = pdf( 2.0, 1, -1.0 ); 107 // returns NaN 108 ``` 109 110 #### pdf.factory( k, lambda ) 111 112 Returns a `function` for evaluating the [PDF][pdf] for an [Erlang][erlang-distribution] distribution with parameters `k` (shape parameter) and `lambda` (rate parameter). 113 114 ```javascript 115 var mypdf = pdf.factory( 3, 1.5 ); 116 117 var y = mypdf( 1.0 ); 118 // returns ~0.377 119 120 y = mypdf( 4.0 ); 121 // returns ~0.067 122 ``` 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 round = require( '@stdlib/math/base/special/round' ); 137 var pdf = require( '@stdlib/stats/base/dists/erlang/pdf' ); 138 139 var lambda; 140 var k; 141 var x; 142 var y; 143 var i; 144 145 for ( i = 0; i < 20; i++ ) { 146 x = randu() * 10.0; 147 k = round( randu() * 10.0 ); 148 lambda = randu() * 5.0; 149 y = pdf( x, k, lambda ); 150 console.log( 'x: %d, k: %d, λ: %d, f(x;k,λ): %d', x.toFixed( 4 ), k, lambda.toFixed( 4 ), y.toFixed( 4 ) ); 151 } 152 ``` 153 154 </section> 155 156 <!-- /.examples --> 157 158 <section class="links"> 159 160 [erlang-distribution]: https://en.wikipedia.org/wiki/Erlang_distribution 161 162 [pdf]: https://en.wikipedia.org/wiki/Probability_density_function 163 164 [degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution 165 166 </section> 167 168 <!-- /.links -->