README.md (4826B)
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 > [Weibull][weibull-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 [Weibull][weibull-distribution] random variable is 30 31 <!-- <equation class="equation" label="eq:weibull_mgf" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \sum_{n=0}^\infty \frac{t^n\lambda^n}{n!}\Gamma\left(1+\frac{n}{k}\right)" alt="Moment-generating function (MGF) for a Weibull distribution."> --> 32 33 <div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \sum_{n=0}^\infty \frac{t^n\lambda^n}{n!}\Gamma\left(1+\frac{n}{k}\right)" data-equation="eq:weibull_mgf"> 34 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/weibull/mgf/docs/img/equation_weibull_mgf.svg" alt="Moment-generating function (MGF) for a Weibull distribution."> 35 <br> 36 </div> 37 38 <!-- </equation> --> 39 40 where `lambda > 0` is the scale paramater and `k > 0` is the shape parameter. 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/weibull/mgf' ); 54 ``` 55 56 #### mgf( t, k, lambda ) 57 58 Evaluates the [moment-generating function][mgf] (MGF) for a [Weibull][weibull-distribution] distribution with [shape parameter][shape] `k` and [scale parameter][scale] `lambda`. 59 60 ```javascript 61 var y = mgf( 1.0, 1.0, 0.5); 62 // returns ~2.0 63 64 y = mgf( -1.0, 4.0, 4.0 ); 65 // returns ~0.019 66 ``` 67 68 If provided `NaN` as any argument, the function returns `NaN`. 69 70 ```javascript 71 var y = mgf( NaN, 1.0, 1.0 ); 72 // returns NaN 73 74 y = mgf( 0.0, NaN, 1.0 ); 75 // returns NaN 76 77 y = mgf( 0.0, 1.0, NaN ); 78 // returns NaN 79 ``` 80 81 If provided `k <= 0`, the function returns `NaN`. 82 83 ```javascript 84 var y = mgf( 0.2, -1.0, 0.5 ); 85 // returns NaN 86 87 y = mgf( 0.2, 0.0, 0.5 ); 88 // returns NaN 89 ``` 90 91 If provided `lambda <= 0`, the function returns `NaN`. 92 93 ```javascript 94 var y = mgf( 0.2, 0.5, -1.0 ); 95 // returns NaN 96 97 y = mgf( 0.2, 0.5, 0.0 ); 98 // returns NaN 99 ``` 100 101 #### mgf.factory( k, lambda ) 102 103 Returns a function for evaluating the [moment-generating function][mgf] of a [Weibull][weibull-distribution] distribution with [shape parameter][shape] `k` and [scale parameter][scale] `lambda`. 104 105 ```javascript 106 var myMGF = mgf.factory( 8.0, 10.0 ); 107 108 var y = myMGF( 0.8 ); 109 // returns ~3150.149 110 111 y = myMGF( 0.08 ); 112 // returns ~2.137 113 ``` 114 115 </section> 116 117 <!-- /.usage --> 118 119 <!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. --> 120 121 <section class="notes"> 122 123 </section> 124 125 <!-- /.notes --> 126 127 <!-- Package usage examples. --> 128 129 <section class="examples"> 130 131 ## Examples 132 133 <!-- eslint no-undef: "error" --> 134 135 ```javascript 136 var randu = require( '@stdlib/random/base/randu' ); 137 var EPS = require( '@stdlib/constants/float64/eps' ); 138 var mgf = require( '@stdlib/stats/base/dists/weibull/mgf' ); 139 140 var lambda; 141 var k; 142 var t; 143 var y; 144 var i; 145 146 for ( i = 0; i < 10; i++ ) { 147 t = randu() * 5.0; 148 lambda = ( randu() * 10.0 ) + EPS; 149 k = ( randu() * 10.0 ) + EPS; 150 y = mgf( t, lambda, k ); 151 console.log( 'x: %d, k: %d, λ: %d, M_X(t;k,λ): %d', t.toFixed( 4 ), k.toFixed( 4 ), lambda.toFixed( 4 ), y.toFixed( 4 ) ); 152 } 153 ``` 154 155 </section> 156 157 <!-- /.examples --> 158 159 <!-- 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. --> 160 161 <section class="references"> 162 163 </section> 164 165 <!-- /.references --> 166 167 <!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. --> 168 169 <section class="links"> 170 171 [weibull-distribution]: https://en.wikipedia.org/wiki/Weibull_distribution 172 173 [mgf]: https://en.wikipedia.org/wiki/Moment-generating_function 174 175 [shape]: https://en.wikipedia.org/wiki/Shape_parameter 176 177 [scale]: https://en.wikipedia.org/wiki/Scale_parameter 178 179 </section> 180 181 <!-- /.links -->