Changeset 744 for trunk/doc


Ignore:
Timestamp:
Feb 10, 2007, 9:16:11 PM (14 years ago)
Author:
Peter
Message:

corrected some typos

File:
1 edited

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  • trunk/doc/Statistics.tex

    r675 r744  
    2222% 02111-1307, USA.
    2323
    24 \usepackage{html}
    2524
    2625
     
    5655{\bf Weighted Statistics}
    5756\normalsize
    58 \begin{htmlonly}
    59 This document is also available in
    60 \htmladdnormallink{PDF}{Statistics.pdf}.
    61 \end{htmlonly}
    6257
    6358\tableofcontents
     
    9691have not chosen one situation for our implementations, so see specific
    9792function documentation for what assumtions are made. Though, common
    98 for implementationare the following:
     93for implementations are the following:
    9994\begin{itemize}
    10095\item Setting all weights to unity yields the same result as the
     
    141136\sigma^2=<X^2>-<X>^2=
    142137\\\frac{\sum w_ix_i^2}{\sum w_i}-\frac{(\sum w_ix_i)^2}{(\sum w_i)^2}=
    143 \\\frac{\sum w_i(x_i^2-m^2)}{\sum w_i}
    144 \\\frac{\sum w_i(x_i^2-2mx_i+m^2)}{\sum w_i}
     138\\\frac{\sum w_i(x_i^2-m^2)}{\sum w_i}=
     139\\\frac{\sum w_i(x_i^2-2mx_i+m^2)}{\sum w_i}=
    145140\\\frac{\sum w_i(x_i-m)^2}{\sum w_i}
    146141\end{eqnarray}
    147142This estimator fulfills that it is invariant under a rescaling and
    148143having a weight equal to zero is equivalent to removing the data
    149 point. Having all weight equal to unity we get $\sigma=\frac{\sum
     144point. Having all weights equal to unity we get $\sigma=\frac{\sum
    150145(x_i-m)^2}{N}$, which is the same as returned from Averager. Hence,
    151146this estimator is slightly biased, but still very efficient.
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