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
|
<HTML>
<HEAD>
<TITLE>Normal Distribution</TITLE>
</HEAD>
<BODY BGCOLOR="#FFFFFF" TEXT="#000000">
<P><font size="+2" color="green">Normal distribution</font></P>
<P>
Assume that each data point, <code>y<sub>k</sub></code>, has an error that is
independently random and distributed as a normal distribution, that is,</P>
<P>
<IMG SRC="img39.gif"></P>
<P>
where <code>σ<sup>2</sup></code> is the variance, and
<code><i>f</i>(x<sub>k</sub>,p)</code> is the expression that we want to fit.</P>
<P>
<IMG SRC="img42.gif"></P>
<P>
The goal is to minimize the <code>χ<sup>2</sup></code> function:</P>
<P>
<IMG SRC="img43.gif"></P>
<P>
where the weights are defined as: <code>w≡1/σ<sup>2</sup></code>.
Consider the Taylor expansion of <code>χ<sup>2</sup></code>:</P>
<P>
<IMG SRC="img46.gif"></P>
<P>
Define the arrays <IMG SRC="img27.gif">, <IMG SRC="img28.gif"> and <IMG SRC="img29.gif">:</P>
<P>
<IMG SRC="img47.gif"></P>
<P>
Linearize and the problem reduces to solving the matrix equation</p>
<p>
<center><IMG SRC="img34.gif"></center></P>
<P>
<a href="chi2andweights.htm"><font size="+1" color="olive">Chi-square and weights</font></a><br />
<a href="hint.htm"><font size="+1" color="olive">Hint for physicists</font></a><br />
<a href="degfree.htm"><font size="+1" color="olive">Degrees of freedom</font></a></P>
<P>
<a href="update.htm"><img src="../shadow_left.gif">
<font size="+1" color="olive">Update after a fit</font></a><br />
<a href="poissondist.htm"><img src="../shadow_right.gif">
<font size="+1" color="olive">Poisson distribution</font></a>
</P>
</BODY>
</HTML>
|
