File: The-Levy-alpha_002dStable-Distributions.html

package info (click to toggle)
gsl-ref-html 2.3-1
  • links: PTS
  • area: non-free
  • in suites: bullseye, buster, sid
  • size: 6,876 kB
  • ctags: 4,574
  • sloc: makefile: 35
file content (98 lines) | stat: -rw-r--r-- 5,000 bytes parent folder | download
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016 The GSL Team.

Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3 or
any later version published by the Free Software Foundation; with the
Invariant Sections being "GNU General Public License" and "Free Software
Needs Free Documentation", the Front-Cover text being "A GNU Manual",
and with the Back-Cover Text being (a) (see below). A copy of the
license is included in the section entitled "GNU Free Documentation
License".

(a) The Back-Cover Text is: "You have the freedom to copy and modify this
GNU Manual." -->
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>GNU Scientific Library &ndash; Reference Manual: The Levy alpha-Stable Distributions</title>

<meta name="description" content="GNU Scientific Library &ndash; Reference Manual: The Levy alpha-Stable Distributions">
<meta name="keywords" content="GNU Scientific Library &ndash; Reference Manual: The Levy alpha-Stable Distributions">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<link href="index.html#Top" rel="start" title="Top">
<link href="Function-Index.html#Function-Index" rel="index" title="Function Index">
<link href="Random-Number-Distributions.html#Random-Number-Distributions" rel="up" title="Random Number Distributions">
<link href="The-Levy-skew-alpha_002dStable-Distribution.html#The-Levy-skew-alpha_002dStable-Distribution" rel="next" title="The Levy skew alpha-Stable Distribution">
<link href="The-Landau-Distribution.html#The-Landau-Distribution" rel="previous" title="The Landau Distribution">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.indentedblock {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
div.smalllisp {margin-left: 3.2em}
kbd {font-style:oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nocodebreak {white-space:nowrap}
span.nolinebreak {white-space:nowrap}
span.roman {font-family:serif; font-weight:normal}
span.sansserif {font-family:sans-serif; font-weight:normal}
ul.no-bullet {list-style: none}
-->
</style>


</head>

<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="The-Levy-alpha_002dStable-Distributions"></a>
<div class="header">
<p>
Next: <a href="The-Levy-skew-alpha_002dStable-Distribution.html#The-Levy-skew-alpha_002dStable-Distribution" accesskey="n" rel="next">The Levy skew alpha-Stable Distribution</a>, Previous: <a href="The-Landau-Distribution.html#The-Landau-Distribution" accesskey="p" rel="previous">The Landau Distribution</a>, Up: <a href="Random-Number-Distributions.html#Random-Number-Distributions" accesskey="u" rel="up">Random Number Distributions</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="The-Levy-alpha_002dStable-Distributions-1"></a>
<h3 class="section">20.13 The Levy alpha-Stable Distributions</h3>
<dl>
<dt><a name="index-gsl_005fran_005flevy"></a>Function: <em>double</em> <strong>gsl_ran_levy</strong> <em>(const gsl_rng * <var>r</var>, double <var>c</var>, double <var>alpha</var>)</em></dt>
<dd><a name="index-Levy-distribution"></a>
<p>This function returns a random variate from the Levy symmetric stable
distribution with scale <var>c</var> and exponent <var>alpha</var>.  The symmetric
stable probability distribution is defined by a Fourier transform,
</p>
<div class="example">
<pre class="example">p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha)
</pre></div>

<p>There is no explicit solution for the form of <em>p(x)</em> and the
library does not define a corresponding <code>pdf</code> function.  For
<em>\alpha = 1</em> the distribution reduces to the Cauchy distribution.  For
<em>\alpha = 2</em> it is a Gaussian distribution with <em>\sigma = \sqrt{2} c</em>.  For <em>\alpha &lt; 1</em> the tails of the
distribution become extremely wide.
</p>
<p>The algorithm only works for <em>0 &lt; alpha &lt;= 2</em>.
</p></dd></dl>

<br>




</body>
</html>