README.md (3602B)
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 > [Chi][chi-distribution] distribution [probability density function][pdf] (PDF). 24 25 <section class="intro"> 26 27 The [probability density function][pdf] (PDF) for a [chi][chi-distribution] random variable is 28 29 <!-- <equation class="equation" label="eq:chi_pdf" align="center" raw="f(x;\,k) = \frac{2^{{1-k/2}}x^{{k-1}}e^{{-x^{2}/2}}}{\Gamma(k/2)}" alt="Probability density function (PDF) for a chi distribution."> --> 30 31 <div class="equation" align="center" data-raw-text="f(x;\,k) = \frac{2^{{1-k/2}}x^{{k-1}}e^{{-x^{2}/2}}}{\Gamma(k/2)}" data-equation="eq:chi_pdf"> 32 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/chi/pdf/docs/img/equation_chi_pdf.svg" alt="Probability density function (PDF) for a chi distribution."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 where `k` is the degrees of freedom and `Γ` denotes the [gamma][gamma-function] function. 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/chi/pdf' ); 50 ``` 51 52 #### pdf( x, k ) 53 54 Evaluates the [probability density function][pdf] (PDF) for a [chi][chi-distribution] distribution with degrees of freedom `k`. 55 56 ```javascript 57 var y = pdf( 0.1, 1.0 ); 58 // returns ~0.794 59 60 y = pdf( 0.5, 2.0 ); 61 // returns ~0.441 62 63 y = pdf( -1.0, 4.0 ); 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, 1.0 ); 71 // returns NaN 72 73 y = pdf( 0.0, NaN ); 74 // returns NaN 75 ``` 76 77 If provided `k < 0`, the function returns `NaN`. 78 79 ```javascript 80 var y = pdf( 2.0, -2.0 ); 81 // returns NaN 82 ``` 83 84 If provided `k = 0`, the function evaluates the [PDF][pdf] of a [degenerate distribution][degenerate-distribution] centered at `0`. 85 86 ```javascript 87 var y = pdf( 2.0, 0.0 ); 88 // returns 0.0 89 90 y = pdf( 0.0, 0.0 ); 91 // returns Infinity 92 ``` 93 94 #### pdf.factory( k ) 95 96 Returns a `function` for evaluating the [PDF][pdf] for a [chi][chi-distribution] distribution with degrees of freedom `k`. 97 98 ```javascript 99 var myPDF = pdf.factory( 6.0 ); 100 101 var y = myPDF( 3.0 ); 102 // returns ~0.337 103 104 y = myPDF( 1.0 ); 105 // returns ~0.076 106 ``` 107 108 </section> 109 110 <!-- /.usage --> 111 112 <section class="examples"> 113 114 ## Examples 115 116 <!-- eslint no-undef: "error" --> 117 118 ```javascript 119 var randu = require( '@stdlib/random/base/randu' ); 120 var pdf = require( '@stdlib/stats/base/dists/chi/pdf' ); 121 122 var k; 123 var x; 124 var y; 125 var i; 126 127 for ( i = 0; i < 20; i++ ) { 128 x = randu() * 10.0; 129 k = randu() * 10.0; 130 y = pdf( x, k ); 131 console.log( 'x: %d, k: %d, f(x;k): %d', x.toFixed( 4 ), k.toFixed( 4 ), y.toFixed( 4 ) ); 132 } 133 ``` 134 135 </section> 136 137 <!-- /.examples --> 138 139 <section class="links"> 140 141 [chi-distribution]: https://en.wikipedia.org/wiki/Chi_distribution 142 143 [degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution 144 145 [gamma-function]: https://en.wikipedia.org/wiki/Gamma_function 146 147 [pdf]: https://en.wikipedia.org/wiki/Probability_density_function 148 149 </section> 150 151 <!-- /.links -->