README.md (3698B)
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 > [Lévy][levy-distribution] distribution [probability density function (PDF)][pdf]. 24 25 <section class="intro"> 26 27 The [probability density function][pdf] (PDF) for a [Lévy][levy-distribution] random variable is 28 29 <!-- <equation class="equation" label="eq:levy_pdf" align="center" raw="f(x;\mu,c)=\begin{cases} \sqrt{\frac{c}{2\pi}}~~\frac{e^{ -\frac{c}{2(x-\mu)}}} {(x-\mu)^{3/2}} & \text{ for } x > \mu \\ 0 & \text{ otherwise} \end{cases}" alt="Probability density function (PDF) for a Lévy distribution."> --> 30 31 <div class="equation" align="center" data-raw-text="f(x;\mu,c)=\begin{cases} \sqrt{\frac{c}{2\pi}}~~\frac{e^{ -\frac{c}{2(x-\mu)}}} {(x-\mu)^{3/2}} & \text{ for } x > \mu \\ 0 & \text{ otherwise} \end{cases}" data-equation="eq:levy_pdf"> 32 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/levy/pdf/docs/img/equation_levy_pdf.svg" alt="Probability density function (PDF) for a Lévy distribution."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 where `μ` is the location parameter and `c > 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/levy/pdf' ); 50 ``` 51 52 #### pdf( x, mu, c ) 53 54 Evaluates the [probability density function][pdf] (PDF) for a [Lévy][levy-distribution] distribution with parameters `mu` (location parameter) and `c` (scale parameter). 55 56 ```javascript 57 var y = pdf( 2.0, 0.0, 1.0 ); 58 // returns ~0.11 59 60 y = pdf( -1.0, 4.0, 4.0 ); 61 // returns 0.0 62 ``` 63 64 If provided `NaN` as any argument, the function returns `NaN`. 65 66 ```javascript 67 var y = pdf( NaN, 0.0, 1.0 ); 68 // returns NaN 69 70 y = pdf( 0.0, NaN, 1.0 ); 71 // returns NaN 72 73 y = pdf( 0.0, 0.0, NaN ); 74 // returns NaN 75 ``` 76 77 If provided `c <= 0`, the function returns `NaN`. 78 79 ```javascript 80 var y = pdf( 2.0, 0.0, -1.0 ); 81 // returns NaN 82 83 y = pdf( 2.0, 0.0, 0.0 ); 84 // returns NaN 85 ``` 86 87 #### pdf.factory( mu, c ) 88 89 Returns a function for evaluating the [probability density function][pdf] (PDF) of a [Lévy][levy-distribution] distribution with parameters `mu` (location parameter) and `c` (scale parameter). 90 91 ```javascript 92 var mypdf = pdf.factory( 10.0, 2.0 ); 93 94 var y = mypdf( 11.0 ); 95 // returns ~0.208 96 97 y = mypdf( 20.0 ); 98 // returns ~0.016 99 ``` 100 101 </section> 102 103 <!-- /.usage --> 104 105 <section class="examples"> 106 107 ## Examples 108 109 <!-- eslint no-undef: "error" --> 110 111 ```javascript 112 var randu = require( '@stdlib/random/base/randu' ); 113 var EPS = require( '@stdlib/constants/float64/eps' ); 114 var pdf = require( '@stdlib/stats/base/dists/levy/pdf' ); 115 116 var mu; 117 var c; 118 var x; 119 var y; 120 var i; 121 122 for ( i = 0; i < 10; i++ ) { 123 mu = randu() * 10.0; 124 x = ( randu()*10.0 ) + mu; 125 c = ( randu()*10.0 ) + EPS; 126 y = pdf( x, mu, c ); 127 console.log( 'x: %d, µ: %d, c: %d, f(x;µ,c): %d', x, mu, c, y ); 128 } 129 ``` 130 131 </section> 132 133 <!-- /.examples --> 134 135 <section class="links"> 136 137 [levy-distribution]: https://en.wikipedia.org/wiki/L%C3%A9vy_distribution 138 139 [pdf]: https://en.wikipedia.org/wiki/Probability_density_function 140 141 </section> 142 143 <!-- /.links -->