README.md (4680B)
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 > [Uniform][uniform-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 [continuous uniform][uniform-distribution] random variable is 30 31 <!-- <equation class="equation" label="eq:uniform_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right]= \begin{cases} \frac{\mathrm{e}^{tb}-\mathrm{e}^{ta}}{t(b-a)} & \text{for } t \neq 0 \\ 1 & \text{for } t = 0 \end{cases}" alt="Moment-generating function (MGF) for a uniform distribution."> --> 32 33 <div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right]= \begin{cases} \frac{\mathrm{e}^{tb}-\mathrm{e}^{ta}}{t(b-a)} & \text{for } t \neq 0 \\ 1 & \text{for } t = 0 \end{cases}" data-equation="eq:uniform_mgf_function"> 34 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/uniform/mgf/docs/img/equation_uniform_mgf_function.svg" alt="Moment-generating function (MGF) for a uniform distribution."> 35 <br> 36 </div> 37 38 <!-- </equation> --> 39 40 where `a` is the minimum support and `b` is the maximum support. The parameters must satisfy `a < b`. 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/uniform/mgf' ); 54 ``` 55 56 #### mgf( t, a, b ) 57 58 Evaluates the [moment-generating function][mgf] (MGF) for a [continuous uniform][uniform-distribution] distribution with parameters `a` (minimum support) and `b` (maximum support). 59 60 ```javascript 61 var y = mgf( 2.0, 0.0, 4.0 ); 62 // returns ~372.495 63 64 y = mgf( -0.2, 0.0, 4.0 ); 65 // returns ~0.688 66 67 y = mgf( 2.0, 0.0, 1.0 ); 68 // returns ~3.195 69 ``` 70 71 If provided `NaN` as any argument, the function returns `NaN`. 72 73 ```javascript 74 var y = mgf( NaN, 0.0, 1.0 ); 75 // returns NaN 76 77 y = mgf( 0.0, NaN, 1.0 ); 78 // returns NaN 79 80 y = mgf( 0.0, 0.0, NaN ); 81 // returns NaN 82 ``` 83 84 If provided `a >= b`, the function returns `NaN`. 85 86 ```javascript 87 var y = mgf( 0.5, 3.0, 2.0 ); 88 // returns NaN 89 90 y = mgf( 0.5, 3.0, 3.0 ); 91 // returns NaN 92 ``` 93 94 #### mgf.factory( a, b ) 95 96 Returns a function for evaluating the [moment-generating function][mgf] (MGF) of a [continuous uniform][uniform-distribution] distribution with parameters `a` (minimum support) and `b` (maximum support). 97 98 ```javascript 99 var mymgf = mgf.factory( 6.0, 7.0 ); 100 var y = mymgf( 0.1 ); 101 // returns ~1.916 102 103 y = mymgf( 1.1 ); 104 // returns ~1339.321 105 ``` 106 107 </section> 108 109 <!-- /.usage --> 110 111 <!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. --> 112 113 <section class="notes"> 114 115 </section> 116 117 <!-- /.notes --> 118 119 <!-- Package usage examples. --> 120 121 <section class="examples"> 122 123 ## Examples 124 125 <!-- eslint no-undef: "error" --> 126 127 ```javascript 128 var randu = require( '@stdlib/random/base/randu' ); 129 var mgf = require( '@stdlib/stats/base/dists/uniform/mgf' ); 130 131 var a; 132 var b; 133 var t; 134 var v; 135 var i; 136 137 for ( i = 0; i < 10; i++ ) { 138 t = randu(); 139 a = randu() * 5.0; 140 b = a + (randu() * 5.0); 141 v = mgf( t, a, b ); 142 console.log( 't: %d, a: %d, b: %d, M_X(t;a,b): %d', t.toFixed( 4 ), a.toFixed( 4 ), b.toFixed( 4 ), v.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 [uniform-distribution]: https://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 163 164 [mgf]: https://en.wikipedia.org/wiki/Moment-generating_function 165 166 </section> 167 168 <!-- /.links -->