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
99
100
101
102
103
104
105
106
|
<!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 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 no
Invariant Sections and no cover texts. A copy of the license is
included in the section entitled "GNU Free Documentation License". -->
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>GNU Scientific Library – Reference Manual: Random Number Distribution Introduction</title>
<meta name="description" content="GNU Scientific Library – Reference Manual: Random Number Distribution Introduction">
<meta name="keywords" content="GNU Scientific Library – Reference Manual: Random Number Distribution Introduction">
<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-Gaussian-Distribution.html#The-Gaussian-Distribution" rel="next" title="The Gaussian Distribution">
<link href="Random-Number-Distributions.html#Random-Number-Distributions" rel="previous" title="Random Number Distributions">
<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="Random-Number-Distribution-Introduction"></a>
<div class="header">
<p>
Next: <a href="The-Gaussian-Distribution.html#The-Gaussian-Distribution" accesskey="n" rel="next">The Gaussian Distribution</a>, Up: <a href="Random-Number-Distributions.html#Random-Number-Distributions" accesskey="u" rel="up">Random Number Distributions</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Introduction-3"></a>
<h3 class="section">20.1 Introduction</h3>
<p>Continuous random number distributions are defined by a probability
density function, <em>p(x)</em>, such that the probability of <em>x</em>
occurring in the infinitesimal range <em>x</em> to <em>x+dx</em> is <em>p dx</em>.
</p>
<p>The cumulative distribution function for the lower tail <em>P(x)</em> is
defined by the integral,
and gives the probability of a variate taking a value less than <em>x</em>.
</p>
<p>The cumulative distribution function for the upper tail <em>Q(x)</em> is
defined by the integral,
and gives the probability of a variate taking a value greater than <em>x</em>.
</p>
<p>The upper and lower cumulative distribution functions are related by
<em>P(x) + Q(x) = 1</em> and satisfy <em>0 <= P(x) <= 1</em>, <em>0 <= Q(x) <= 1</em>.
</p>
<p>The inverse cumulative distributions, <em>x=P^{-1}(P)</em> and <em>x=Q^{-1}(Q)</em> give the values of <em>x</em>
which correspond to a specific value of <em>P</em> or <em>Q</em>.
They can be used to find confidence limits from probability values.
</p>
<p>For discrete distributions the probability of sampling the integer
value <em>k</em> is given by <em>p(k)</em>, where <em>\sum_k p(k) = 1</em>.
The cumulative distribution for the lower tail <em>P(k)</em> of a
discrete distribution is defined as,
where the sum is over the allowed range of the distribution less than
or equal to <em>k</em>.
</p>
<p>The cumulative distribution for the upper tail of a discrete
distribution <em>Q(k)</em> is defined as
giving the sum of probabilities for all values greater than <em>k</em>.
These two definitions satisfy the identity <em>P(k)+Q(k)=1</em>.
</p>
<p>If the range of the distribution is 1 to <em>n</em> inclusive then
<em>P(n)=1</em>, <em>Q(n)=0</em> while <em>P(1) = p(1)</em>,
<em>Q(1)=1-p(1)</em>.
</p>
</body>
</html>
|
