README.md (3732B)
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 > [Degenerate distribution][degenerate-distribution] [probability density function][pdf] (PDF). 24 25 <section class="intro"> 26 27 Strictly speaking, as a discrete distribution, a [degenerate][degenerate-distribution] has no [probability density function][pdf] (PDF). Extending the notion of a [PDF][pdf], we conceptualize the [PDF][pdf] of a [degenerate][degenerate-distribution] as an infinitely tall spike centered at `mu`. More formally, 28 29 <!-- <equation class="equation" label="eq:degenerate_pdf" align="center" raw="f(x;\mu) = \delta(x-\mu)" alt="Probability density function (PDF) for a degenerate distribution."> --> 30 31 <div class="equation" align="center" data-raw-text="f(x;\mu) = \delta(x-\mu)" data-equation="eq:degenerate_pdf"> 32 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/degenerate/pdf/docs/img/equation_degenerate_pdf.svg" alt="Probability density function (PDF) for a degenerate distribution."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 where `delta` is the Dirac delta function. 39 40 <!-- <equation class="equation" label="eq:dirac_delta" align="center" raw="\delta(x) = {\begin{cases}+\infty, & x = 0\\0, & x \neq 0\end{cases}}" alt="Dirac delta function."> --> 41 42 <div class="equation" align="center" data-raw-text="\delta(x) = {\begin{cases}+\infty, & x = 0\\0, & x \neq 0\end{cases}}" data-equation="eq:dirac_delta"> 43 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/degenerate/pdf/docs/img/equation_dirac_delta.svg" alt="Dirac delta function."> 44 <br> 45 </div> 46 47 <!-- </equation> --> 48 49 </section> 50 51 <!-- /.intro --> 52 53 <section class="usage"> 54 55 ## Usage 56 57 ```javascript 58 var pdf = require( '@stdlib/stats/base/dists/degenerate/pdf' ); 59 ``` 60 61 #### pdf( x, mu ) 62 63 Evaluates the [PDF][pdf] of a [degenerate distribution][degenerate-distribution] centered at `mu`. 64 65 ```javascript 66 var y = pdf( 2.0, 8.0 ); 67 // returns 0.0 68 69 y = pdf( 8.0, 8.0 ); 70 // returns Infinity 71 ``` 72 73 #### pdf.factory( mu ) 74 75 Returns a function for evaluating the [PDF][pdf] of a [degenerate distribution][degenerate-distribution] centered at `mu`. 76 77 ```javascript 78 var mypdf = pdf.factory( 10.0 ); 79 80 var y = mypdf( 10.0 ); 81 // returns Infinity 82 83 y = mypdf( 5.0 ); 84 // returns 0.0 85 86 y = mypdf( 12.0 ); 87 // returns 0.0 88 ``` 89 90 </section> 91 92 <!-- /.usage --> 93 94 <section class="examples"> 95 96 ## Examples 97 98 <!-- eslint no-undef: "error" --> 99 100 ```javascript 101 var randu = require( '@stdlib/random/base/randu' ); 102 var round = require( '@stdlib/math/base/special/round' ); 103 var pdf = require( '@stdlib/stats/base/dists/degenerate/pdf' ); 104 105 var mu; 106 var x; 107 var y; 108 var i; 109 110 for ( i = 0; i < 100; i++ ) { 111 x = round( randu()*5.0 ); 112 mu = round( randu()*5.0 ); 113 y = pdf( x, mu ); 114 console.log( 'x: %d, µ: %d, f(x;µ): %d', x, mu, y ); 115 } 116 ``` 117 118 </section> 119 120 <!-- /.examples --> 121 122 <section class="links"> 123 124 [pdf]: https://en.wikipedia.org/wiki/Probability_density_function 125 126 [degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution 127 128 </section> 129 130 <!-- /.links -->