README.md (4313B)
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 # Moment-Generating Function 22 23 > [Chi-squared][chisquare-distribution] distribution moment-generating function (MGF). 24 25 <!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. --> 26 27 <section class="intro"> 28 29 The [moment-generating function][mgf] for a [chi-squared][chisquare-distribution] random variable is 30 31 <!-- <equation class="equation" label="eq:chisquare_mgf" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \left( 1 - 2t \right )^{-k/2} \text{ for } t < \tfrac{1}{2}" alt="Moment-generating function (MGF) for a chi-squared distribution."> --> 32 33 <div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \left( 1 - 2t \right )^{-k/2} \text{ for } t < \tfrac{1}{2}" data-equation="eq:chisquare_mgf"> 34 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/chisquare/mgf/docs/img/equation_chisquare_mgf.svg" alt="Moment-generating function (MGF) for a chi-squared distribution."> 35 <br> 36 </div> 37 38 <!-- </equation> --> 39 40 where `k` is the degrees of freedom. 41 42 </section> 43 44 <!-- /.intro --> 45 46 <!-- Package usage documentation. --> 47 48 <section class="usage"> 49 50 ## Usage 51 52 ```javascript 53 var mgf = require( '@stdlib/stats/base/dists/chisquare/mgf' ); 54 ``` 55 56 #### mgf( t, k ) 57 58 Evaluates the [moment-generating function][mgf] (MGF) for a [chi-squared][chisquare-distribution] distribution with degrees of freedom `k`. 59 60 ```javascript 61 var y = mgf( 0.4, 2 ); 62 // returns ~5.0 63 64 y = mgf( -1.0, 5.0 ); 65 // returns ~0.0642 66 67 y = mgf( 0.0, 10.0 ); 68 // returns 1.0 69 ``` 70 71 If provided `NaN` as any argument, the function returns `NaN`. 72 73 ```javascript 74 var y = mgf( NaN, 1.0 ); 75 // returns NaN 76 77 y = mgf( 0.0, NaN ); 78 // returns NaN 79 ``` 80 81 If provided `t >= 0.5`, the function returns `NaN`. 82 83 ```javascript 84 var y = mgf( 0.8, 1.0 ); 85 // returns NaN 86 ``` 87 88 If provided `k < 0`, the function returns `NaN`. 89 90 ```javascript 91 var y = mgf( 2.0, -2.0 ); 92 // returns NaN 93 ``` 94 95 #### mgf.factory( k ) 96 97 Returns a function for evaluating the [moment-generating function][mgf] (MGF) for a [chi-squared][chisquare-distribution] distribution with degrees of freedom `k`. 98 99 ```javascript 100 var mymgf = mgf.factory( 1.0 ); 101 102 var y = mymgf( 0.2 ); 103 // returns ~1.291 104 105 y = mymgf( 0.4 ); 106 // returns ~2.236 107 ``` 108 109 </section> 110 111 <!-- /.usage --> 112 113 <!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. --> 114 115 <section class="notes"> 116 117 </section> 118 119 <!-- /.notes --> 120 121 <!-- Package usage examples. --> 122 123 <section class="examples"> 124 125 ## Examples 126 127 <!-- eslint no-undef: "error" --> 128 129 ```javascript 130 var randu = require( '@stdlib/random/base/randu' ); 131 var mgf = require( '@stdlib/stats/base/dists/chisquare/mgf' ); 132 133 var t; 134 var k; 135 var y; 136 var i; 137 138 for ( i = 0; i < 10; i++ ) { 139 t = randu() * 0.5; 140 k = randu() * 10.0; 141 y = mgf( t, k ); 142 console.log( 'x: %d, k: %d, M_X(t;k): %d', t.toFixed( 4 ), k.toFixed( 4 ), y.toFixed( 4 ) ); 143 } 144 ``` 145 146 </section> 147 148 <!-- /.examples --> 149 150 <!-- Section to include cited references. If references are included, add a horizontal rule *before* the section. Make sure to keep an empty line after the `section` element and another before the `/section` close. --> 151 152 <section class="references"> 153 154 </section> 155 156 <!-- /.references --> 157 158 <!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. --> 159 160 <section class="links"> 161 162 [chisquare-distribution]: https://en.wikipedia.org/wiki/Chi-squared_distribution 163 164 [mgf]: https://en.wikipedia.org/wiki/Moment-generating_function 165 166 </section> 167 168 <!-- /.links -->