README.md (3318B)
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 > [Student's t][t-distribution] distribution probability density function (PDF). 24 25 <section class="intro"> 26 27 The [probability density function][pdf] (PDF) for a [t distribution][t-distribution] random variable is 28 29 <!-- <equation class="equation" label="eq:t_pdf" align="center" raw="\frac{1} {\sqrt{\nu}\,B\left( \tfrac{1}{2}, \tfrac{\nu}{2} \right )} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}" alt="Probability density function (PDF) for a Student's t distribution."> --> 30 31 <div class="equation" align="center" data-raw-text="\frac{1} {\sqrt{\nu}\,B\left( \tfrac{1}{2}, \tfrac{\nu}{2} \right )} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}" data-equation="eq:t_pdf"> 32 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/t/pdf/docs/img/equation_t_pdf.svg" alt="Probability density function (PDF) for a Student's t distribution."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 where `v > 0` is the degrees of freedom. 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/t/pdf' ); 50 ``` 51 52 #### pdf( x, v ) 53 54 Evaluates the [probability density function][pdf] (PDF) for a [Student's t][t-distribution] distribution with degrees of freedom `v`. 55 56 ```javascript 57 var y = pdf( 0.3, 4.0 ); 58 // returns ~0.355 59 60 y = pdf( 2.0, 0.7 ); 61 // returns ~0.058 62 63 y = pdf( -1.0, 0.5 ); 64 // returns ~0.118 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 `v <= 0`, the function returns `NaN`. 78 79 ```javascript 80 var y = pdf( 2.0, -1.0 ); 81 // returns NaN 82 83 y = pdf( 2.0, 0.0 ); 84 // returns NaN 85 ``` 86 87 #### pdf.factory( v ) 88 89 Returns a `function` for evaluating the [PDF][pdf] of a [Student's t][t-distribution] distribution with degrees of freedom `v`. 90 91 ```javascript 92 var mypdf = pdf.factory( 1.0 ); 93 var y = mypdf( 3.0 ); 94 // returns ~0.032 95 96 y = mypdf( 1.0 ); 97 // returns ~0.159 98 ``` 99 100 </section> 101 102 <!-- /.usage --> 103 104 <section class="examples"> 105 106 ## Examples 107 108 <!-- eslint no-undef: "error" --> 109 110 ```javascript 111 var randu = require( '@stdlib/random/base/randu' ); 112 var pdf = require( '@stdlib/stats/base/dists/t/pdf' ); 113 114 var v; 115 var x; 116 var y; 117 var i; 118 119 for ( i = 0; i < 10; i++ ) { 120 x = (randu() * 6.0) - 3.0; 121 v = randu() * 10.0; 122 y = pdf( x, v ); 123 console.log( 'x: %d, v: %d, f(x;v): %d', x, v, y ); 124 } 125 ``` 126 127 </section> 128 129 <!-- /.examples --> 130 131 <section class="links"> 132 133 [pdf]: https://en.wikipedia.org/wiki/Probability_density_function 134 135 [t-distribution]: https://en.wikipedia.org/wiki/Student%27s_t-distribution 136 137 </section> 138 139 <!-- /.links -->