README.md (4971B)
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 > [Triangular][triangular-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 [triangular][triangular-distribution] random variable is 30 31 <!-- <equation class="equation" label="eq:triangular_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = 2\frac{(b\!-\!c)e^{at}\!-\!(b\!-\!a)e^{ct}\!+\!(c\!-\!a)e^{bt}} {(b-a)(c-a)(b-c)t^2}" alt="Moment-generating function (MGF) for a triangular distribution."> --> 32 33 <div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = 2\frac{(b\!-\!c)e^{at}\!-\!(b\!-\!a)e^{ct}\!+\!(c\!-\!a)e^{bt}} {(b-a)(c-a)(b-c)t^2}" data-equation="eq:triangular_mgf_function"> 34 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/docs/img/equation_triangular_mgf_function.svg" alt="Moment-generating function (MGF) for a triangular distribution."> 35 <br> 36 </div> 37 38 <!-- </equation> --> 39 40 where `a` is the lower limit, `b` is the upper limit, and `c` is the mode of the distribution. The parameters must satisfy `b > a` and `a <= b <= c`. 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/triangular/mgf' ); 54 ``` 55 56 #### mgf( t, a, b, c ) 57 58 Evaluates the [moment-generating function][mgf] (MGF) for a [triangular][triangular-distribution] distribution with parameters `a` (lower limit), `b` (upper limit), and `c` (mode). 59 60 ```javascript 61 var y = mgf( 0.5, -1.0, 1.0, 0.0 ); 62 // returns ~1.021 63 64 y = mgf( 0.5, -1.0, 1.0, 0.5 ); 65 // returns ~1.111 66 67 y = mgf( -0.3, -20.0, 0.0, -2.0 ); 68 // returns ~24.334 69 70 y = mgf( -2.0, -1.0, 1.0, 0.0 ); 71 // returns ~1.381 72 ``` 73 74 If provided `NaN` as any argument, the function returns `NaN`. 75 76 ```javascript 77 var y = mgf( NaN, 0.0, 1.0, 0.5 ); 78 // returns NaN 79 80 y = mgf( 0.0, NaN, 1.0, 0.5 ); 81 // returns NaN 82 83 y = mgf( 0.0, 0.0, NaN, 0.5 ); 84 // returns NaN 85 86 y = mgf( 2.0, 1.0, 0.0, NaN ); 87 // returns NaN 88 ``` 89 90 If provided parameters not satisfying `a <= c <= b`, the function returns `NaN`. 91 92 ```javascript 93 var y = mgf( 2.0, 1.0, 0.0, 1.5 ); 94 // returns NaN 95 96 y = mgf( 2.0, 1.0, 0.0, -1.0 ); 97 // returns NaN 98 99 y = mgf( 2.0, 0.0, -1.0, 0.5 ); 100 // returns NaN 101 ``` 102 103 #### mgf.factory( a, b, c ) 104 105 Returns a function for evaluating the [moment-generating function][mgf] of a [triangular][triangular-distribution] distribution with parameters `a` (lower limit), `b` (upper limit), and `c` (mode). 106 107 ```javascript 108 var mymgf = mgf.factory( 0.0, 2.0, 1.0 ); 109 110 var y = mymgf( -1.0 ); 111 // returns ~0.3996 112 113 y = mymgf( 2.0 ); 114 // returns ~10.205 115 ``` 116 117 </section> 118 119 <!-- /.usage --> 120 121 <!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. --> 122 123 <section class="notes"> 124 125 </section> 126 127 <!-- /.notes --> 128 129 <!-- Package usage examples. --> 130 131 <section class="examples"> 132 133 ## Examples 134 135 <!-- eslint no-undef: "error" --> 136 137 ```javascript 138 var randu = require( '@stdlib/random/base/randu' ); 139 var mgf = require( '@stdlib/stats/base/dists/triangular/mgf' ); 140 141 var a; 142 var b; 143 var c; 144 var t; 145 var v; 146 var i; 147 148 for ( i = 0; i < 10; i++ ) { 149 t = randu() * 5.0; 150 a = randu() * 10.0; 151 b = a + (randu() * 40.0); 152 c = a + (( b - a ) * randu()); 153 v = mgf( t, a, b, c ); 154 console.log( 't: %d, a: %d, b: %d, c: %d, M_X(t;a,b,c): %d', t.toFixed( 4 ), a.toFixed( 4 ), b.toFixed( 4 ), c.toFixed( 4 ), v.toFixed( 4 ) ); 155 } 156 ``` 157 158 </section> 159 160 <!-- /.examples --> 161 162 <!-- 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. --> 163 164 <section class="references"> 165 166 </section> 167 168 <!-- /.references --> 169 170 <!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. --> 171 172 <section class="links"> 173 174 [triangular-distribution]: https://en.wikipedia.org/wiki/Triangular_distribution 175 176 [mgf]: https://en.wikipedia.org/wiki/Moment-generating_function 177 178 </section> 179 180 <!-- /.links -->