README.md (4416B)
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 > [Normal][normal-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 [normal][normal-distribution] random variable is 30 31 <!-- <equation class="equation" label="eq:normal_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \exp\{ \mu t + \frac{1}{2}\sigma^2t^2 \}" alt="Moment-generating function (MGF) for a normal distribution."> --> 32 33 <div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \exp\{ \mu t + \frac{1}{2}\sigma^2t^2 \}" data-equation="eq:normal_mgf_function"> 34 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/normal/mgf/docs/img/equation_normal_mgf_function.svg" alt="Moment-generating function (MGF) for a normal distribution."> 35 <br> 36 </div> 37 38 <!-- </equation> --> 39 40 where `mu` is the mean and `sigma > 0` is the standard deviation. 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/normal/mgf' ); 54 ``` 55 56 #### mgf( t, mu, sigma ) 57 58 Evaluates the [moment-generating function][mgf] (MGF) for a [normal][normal-distribution] distribution with parameters `mu` (mean) and `sigma` (standard deviation). 59 60 ```javascript 61 var y = mgf( 2.0, 0.0, 1.0 ); 62 // returns ~7.389 63 64 y = mgf( 0.0, 0.0, 1.0 ); 65 // returns 1.0 66 67 y = mgf( -1.0, 4.0, 2.0 ); 68 // returns ~0.1353 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 `sigma <= 0`, the function returns `NaN`. 85 86 ```javascript 87 var y = mgf( 2.0, 0.0, 0.0 ); 88 // returns NaN 89 90 y = mgf( 2.0, 0.0, -1.0 ); 91 // returns NaN 92 ``` 93 94 #### mgf.factory( mu, sigma ) 95 96 Returns a function for evaluating the [moment-generating function][mgf] (MGF) of a [normal][normal-distribution] distribution with parameters `mu` and `sigma`. 97 98 ```javascript 99 var mymgf = mgf.factory( 4.0, 2.0 ); 100 101 var y = mymgf( 1.0 ); 102 // returns ~403.429 103 104 y = mymgf( 0.5 ); 105 // returns ~12.182 106 ``` 107 108 </section> 109 110 <!-- /.usage --> 111 112 <!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. --> 113 114 <section class="notes"> 115 116 </section> 117 118 <!-- /.notes --> 119 120 <!-- Package usage examples. --> 121 122 <section class="examples"> 123 124 ## Examples 125 126 <!-- eslint no-undef: "error" --> 127 128 ```javascript 129 var randu = require( '@stdlib/random/base/randu' ); 130 var mgf = require( '@stdlib/stats/base/dists/normal/mgf' ); 131 132 var sigma; 133 var mu; 134 var t; 135 var y; 136 var i; 137 138 for ( i = 0; i < 10; i++ ) { 139 t = randu(); 140 mu = (randu() * 10.0) - 5.0; 141 sigma = randu() * 20.0; 142 y = mgf( t, mu, sigma ); 143 console.log( 't: %d, µ: %d, σ: %d, M_X(t;µ,σ): %d', t.toFixed( 4 ), mu.toFixed( 4 ), sigma.toFixed( 4 ), y.toFixed( 4 ) ); 144 } 145 ``` 146 147 </section> 148 149 <!-- /.examples --> 150 151 <!-- 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. --> 152 153 <section class="references"> 154 155 </section> 156 157 <!-- /.references --> 158 159 <!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. --> 160 161 <section class="links"> 162 163 [normal-distribution]: https://en.wikipedia.org/wiki/Normal_distribution 164 165 [mgf]: https://en.wikipedia.org/wiki/Moment-generating_function 166 167 </section> 168 169 <!-- /.links -->