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slsDetectorPackage/manual/docs/html/slsDetectors-FAQ/images.tex

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\batchmode
\documentclass{report}
\RequirePackage{ifthen}
\usepackage{amssymb}
\usepackage[dvips]{graphicx}
\usepackage{verbatim}
\usepackage{html}
\usepackage{amsmath}
\usepackage{latexsym,amssymb}
\usepackage[mathscr]{eucal}
\usepackage{amsthm,amsxtra,amscd,upref}
\usepackage{layout,bm,dcolumn}
\usepackage{graphicx,color}
\usepackage{calc}
\usepackage{framed}
%
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%
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%
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% \\
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\\
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$ k_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8003}%
$\displaystyle \alpha_{jm}=\Phi_m-{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}}\right)}}\arctan{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{D_m-pj}}}}{{\ensuremath{\displaystyle{R_m}}}}}}}}\right)}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8005}%
$\displaystyle \alpha_{jm}=o_m+{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}}\right)}}c_mk_m+{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}}\right)}}\arctan{\ensuremath{\left[{{\ensuremath{\left({j-c_m}\right)}}k_m}\right]}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8008}%
$\displaystyle c_m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8010}%
$\displaystyle =$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8012}%
$\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{D_m}}}}{{\ensuremath{\displaystyle{p}}}}}}};$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8014}%
$\displaystyle k_m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8018}%
$\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{p}}}}{{\ensuremath{\displaystyle{R_m}}}}}}};$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8020}%
$\displaystyle o_m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8024}%
$\displaystyle \Phi_m-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{D_m}}}}{{\ensuremath{\displaystyle{R_m}}}}}}}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8027}%
$\displaystyle \Phi_m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8031}%
$\displaystyle o_m+{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}c_mk_m;$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8033}%
$\displaystyle R_m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8037}%
$\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{p}}}}{{\ensuremath{\displaystyle{k_m}}}}}}};$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8039}%
$\displaystyle D_m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8043}%
$\displaystyle c_m p.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{section}
\stepcounter{section}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8051}%
$ {\ensuremath{{2\theta}}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8053}%
$ 2\theta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8057}%
$\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{d}{}\, }}\bf {\sigma}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{d}{}\, }}\Omega}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8059}%
$ \Omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8061}%
$\displaystyle {I_0}\Delta t \Delta\Omega{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{d}{}\, }}\bf {\sigma}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{d}{}\, }}\Omega}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8063}%
$ \Delta t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8065}%
$ \Delta\Omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8067}%
$ I_0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8071}%
$ \Delta\Omega\propto \Delta {\ensuremath{{2\theta}}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8083}%
$ P$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8085}%
$ k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8087}%
$ k=1,\ldots,P$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8089}%
$ N_k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8091}%
$ 2\theta\equiv{\ensuremath{{2\theta}}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8093}%
$\displaystyle b_{k,j}={\ensuremath{\left[{{\ensuremath{{2\theta}}}_{k,j}^{-},{\ensuremath{{2\theta}}}_{k,j}^{+}}\right]}},\qquad j=1,\ldots,N_k
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8095}%
$\displaystyle \hat{b}_{k,j}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{{2\theta}}}_{k,j}^{+}+{\ensuremath{{2\theta}}}_{k,j}^{-}}}}}{{\ensuremath{\displaystyle{2}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8097}%
$\displaystyle {\ensuremath{\left|{b_{k,j}}\right|}}={\ensuremath{{2\theta}}}_{k,j}^{+}-{\ensuremath{{2\theta}}}_{k,j}^{-}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8099}%
$ C_{k,j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8101}%
$ e_{k,j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8103}%
$ m_{k,j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8105}%
$ b_{k,j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8107}%
$\displaystyle I_{k,j}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e_{k,j}}}}}{{\ensuremath{\displaystyle{m_{k,j}}}}}}}}{\ensuremath{\left({C_{k,j}+\min{\ensuremath{\left({1,C_{k,j}}\right)}}}\right)}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8109}%
$\displaystyle \sigma_{I_{k,j}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e_{k,j}}}}}{{\ensuremath{\displaystyle{m_{k,j}}}}}}}}\sqrt{{\ensuremath{\left({C_{k,j}+1}\right)}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8111}%
$\displaystyle r_{k,j}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{I_{k,j}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{b_{k,j}}\right|}}}}}}}}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e_{k,j}}}}}{{\ensuremath{\displaystyle{m_{k,j}{\ensuremath{\left|{b_{k,j}}\right|}}}}}}}}}{\ensuremath{\left({C_{k,j}+\min{\ensuremath{\left({1,C_{k,j}}\right)}}}\right)}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8113}%
$\displaystyle \sigma_{r_{k,j}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_{I_{k,j}}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{b_{k,j}}\right|}}}}}}}}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e_{k,j}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{b_{k,j}}\right|}}m_{k,j}}}}}}}}\sqrt{{\ensuremath{\left({C_{k,j}+1}\right)}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8115}%
$ M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8117}%
$\displaystyle B_\ell=[{\ensuremath{{2\theta}}}_0+(\ell-1)B, {\ensuremath{{2\theta}}}_0+\ell B],\qquad \ell=1,\ldots,M
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8119}%
$\displaystyle {\ensuremath{\left|{B_\ell}\right|}}=B$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8121}%
$\displaystyle \hat{B}_\ell={\ensuremath{{2\theta}}}_0+(\ell-1/2)B,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8123}%
$ {\ensuremath{{2\theta}}}_0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8125}%
$ {\ensuremath{{2\theta}}}_{max}={\ensuremath{{2\theta}}}_0+MB$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8127}%
$ \ell$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8129}%
$\displaystyle b_{k,j}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8130}%
$\displaystyle \qquad {\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}} > 0.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8133}%
$\displaystyle \qquad \hat{b}_{k,j}\in B_\ell .
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8137}%
$ B_\ell$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8139}%
$ N_E$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8141}%
$ O_n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8143}%
$ O$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8145}%
$ \sigma_{O_n}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8147}%
$ \nu_n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8149}%
$\displaystyle \langle O\rangle ={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
\mathop{\sum}_{n=1}^{N_E}\nu_n
O_n\sigma_{O_n}^{-2}
}}}}{{\ensuremath{\displaystyle{
\mathop{\sum}_{n=1}^{N_E}\nu_n
\sigma_{O_n}^{-2}
}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8151}%
$\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8153}%
$ k,j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8157}%
$\displaystyle X_\ell=\mathop{\sum_{k,j}}_{ {\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}} > 0}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}\ r_{k,j}\ {\ensuremath{\left({\sigma_{r_{k,j}}}\right)}}^{-2}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8159}%
$\displaystyle Y_\ell=\mathop{\sum_{k,j}}_{ {\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}} > 0}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}\ {\ensuremath{\left({\sigma_{r_{k,j}}}\right)}}^{-2}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8161}%
$\displaystyle R_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{X_\ell}}}}{{\ensuremath{\displaystyle{Y_\ell}}}}}}};
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8163}%
$\displaystyle \sigma_{R_\ell}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{\sqrt{Y_\ell}}}}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8165}%
$ R_\ell$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8167}%
$ \sigma_{R_\ell}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8169}%
$ B$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8171}%
$ K$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8175}%
$\displaystyle \mathop{\sum}_{\ell=1}^M{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{KR_\ell}}}}{{\ensuremath{\displaystyle{K^2\sigma_{R_\ell}^2}}}}}}}=
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{K}}}}}}}
\mathop{\sum}_{\ell=1}^M{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{R_\ell}}}}{{\ensuremath{\displaystyle{\sigma_{R_\ell}^2}}}}}}}=M
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8179}%
$\displaystyle K={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
1
}}}}{{\ensuremath{\displaystyle{
M
}}}}}}}\mathop{\sum}_{\ell=1}^M{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{R_\ell}}}}{{\ensuremath{\displaystyle{\sigma_{R_\ell}^2}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8181}%
$\displaystyle \hat{B}_\ell, \quad KR_\ell, \quad K\sigma_{R_\ell}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8193}%
$ X_\ell=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8195}%
$ Y_\ell=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8197}%
$ b$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8201}%
$\displaystyle X_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}\
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e(C+1)}}}}{{\ensuremath{\displaystyle{m|b|}}}}}}}\
{\ensuremath{\left({
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{|b|m}}}}{{\ensuremath{\displaystyle{e\sqrt{C+1}}}}}}}}
}\right)}}^{2}
={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{|b|m}}}}{{\ensuremath{\displaystyle{e}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8203}%
$\displaystyle Y_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}\
{\ensuremath{\left({
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{|b|m}}}}{{\ensuremath{\displaystyle{e\sqrt{C+1}}}}}}}}
}\right)}}^{2}
={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{|b|^2m^2}}}}{{\ensuremath{\displaystyle{e^2(C+1)}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8205}%
$\displaystyle R_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{X_\ell}}}}{{\ensuremath{\displaystyle{Y_\ell}}}}}}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e(C+1)}}}}{{\ensuremath{\displaystyle{m|b|}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8209}%
$\displaystyle \sigma_{R_\ell}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{\sqrt{Y_\ell}}}}}}}}=
\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{B}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}}}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e\sqrt{(C+1)}}}}}{{\ensuremath{\displaystyle{|b|m}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8213}%
$\displaystyle \sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{B}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8217}%
$ \hat{b}_{j,k}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8219}%
$ \hat{B}_\ell$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8223}%
$ C_0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8227}%
$ \sqrt{C_0}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8229}%
$ n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8231}%
$\displaystyle P(n)={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{C_0^n{\ensuremath{\mathrm{e}}}^{-C_0}
}}}}{{\ensuremath{\displaystyle{
n!}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8233}%
$\displaystyle \mathop{\sum}_{n=0}^{+\infty}
P(n)=1\ ;
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8235}%
$\displaystyle \langle n\rangle=\mathop{\sum}_{n=0}^{+\infty}
nP(n)=C_0\ ;
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8237}%
$\displaystyle \langle n^2\rangle=\mathop{\sum}_{n=0}^{+\infty}
n^2 P(n)=C_0^2+C_0\ ;
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8239}%
$\displaystyle \sigma_{C_0}=\sqrt{\langle n^2\rangle-\langle n\rangle^2}=\sqrt{C_0}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8243}%
$\displaystyle \chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({F_j-O_j}\right)}}^2
}}}}{{\ensuremath{\displaystyle{
\sigma_j^2
}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8245}%
$ O_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8247}%
$ F_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8249}%
$ \sigma_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8251}%
$\displaystyle \chi_{(0)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({F_j-C_j}\right)}}^2
}}}}{{\ensuremath{\displaystyle{
C_j
}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8253}%
$\displaystyle \chi_{(1)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({F_j-{\ensuremath{\left({C_j+\min{\ensuremath{\left({1,C_j}\right)}}}\right)}}}\right)}}^2
}}}}{{\ensuremath{\displaystyle{
C_j+1
}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8261}%
$ N_{\mathrm{obs}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8263}%
$ C_j,\quad j=1\ldots N_{\mathrm{obs}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8265}%
$ x$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8269}%
$\displaystyle x=\langle x\rangle={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}}
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j\ .
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8271}%
$\displaystyle \sigma_x=\sqrt{\langle x^2\rangle-\langle x\rangle^2}=\sqrt{
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}}
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j^2-{\ensuremath{\left({
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}}
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j
}\right)}}
}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8273}%
$\displaystyle \sigma_x={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}}\sqrt{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j }
=\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x\rangle}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8277}%
$ C_j=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8279}%
$ N_{\mathrm{obs}}^*$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8281}%
$\displaystyle x=\langle x\rangle^*={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}^*}}}}}}}
\mathop{\sum}_ {\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
C_j={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}^*}}}}}}}
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}\langle x\rangle
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8283}%
$\displaystyle \sigma_{x^*}= {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}\sigma_x = \sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}}
\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x\rangle}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}}=\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x\rangle^*}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8285}%
$ C_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8295}%
$\displaystyle \chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({x-O_j}\right)}}^2
}}}}{{\ensuremath{\displaystyle{
\sigma_j^2
}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8297}%
$\displaystyle x= \langle x \rangle_{\!\mathrm{w}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{O_j
}}}}{{\ensuremath{\displaystyle{
\sigma_j^2
}}}}}}}
}}}}{{\ensuremath{\displaystyle{
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
}}}}{{\ensuremath{\displaystyle{
\sigma_j^2
}}}}}}}
}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8301}%
$\displaystyle \sigma_{\langle x \rangle_{\!\mathrm{w}}} = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
1
}}}}{{\ensuremath{\displaystyle{\sqrt{
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
}}}}{{\ensuremath{\displaystyle{
\sigma_j^2
}}}}}}}
}}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8303}%
$\displaystyle \mathsf{GoF}=
\sqrt{
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{O_j^2
}}}}{{\ensuremath{\displaystyle{
\sigma_j^2
}}}}}}}
-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
{\ensuremath{\left[{
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{O_j
}}}}{{\ensuremath{\displaystyle{
\sigma_j^2
}}}}}}}
}\right]}}^2
}}}}{{\ensuremath{\displaystyle{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
}}}}{{\ensuremath{\displaystyle{
\sigma_j^2
}}}}}}} }}}}}}}
}}}}{{\ensuremath{\displaystyle{
N_{\mathrm{obs}}-1
}}}}}}}
}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8305}%
$\displaystyle {\sigma}_{\langle x \rangle_{\!\mathrm{w}}}^{\mathrm{corrected}} = \mathsf{GoF}\ \sigma_{\langle x \rangle_{\!\mathrm{w}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8309}%
$ O_j=C_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8311}%
$ \sigma_j^2=C_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8313}%
$\displaystyle \langle x \rangle_{\!\mathrm{w(1)}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
{N_{\mathrm{obs}}}
}}}}{{\ensuremath{\displaystyle{
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
}}}}{{\ensuremath{\displaystyle{
C_j
}}}}}}}
}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8321}%
$\displaystyle \sigma_{\langle x \rangle_{\!\mathrm{w(1)}}} = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
1
}}}}{{\ensuremath{\displaystyle{\sqrt{
\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
}}}}{{\ensuremath{\displaystyle{
C_j
}}}}}}}
}}}}}}}}=\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x \rangle_{\!\mathrm{w(1)}}}}}}{{\ensuremath{\displaystyle{
N_{\mathrm{obs}}^*
}}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8323}%
$\displaystyle \mathsf{GoF}_{(1)}=
\sqrt{
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
\!\!\!\!C_j
-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
{\ensuremath{\left[{
N_{\mathrm{obs}}^*
}\right]}}^2
}}}}{{\ensuremath{\displaystyle{ \mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
}}}}{{\ensuremath{\displaystyle{
C_j
}}}}}}} }}}}}}}
}}}}{{\ensuremath{\displaystyle{
N_{\mathrm{obs}}^*-1
}}}}}}}
}
=\sqrt{
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*-1}}}}}}}
{\ensuremath{\left({
\langle x\rangle^*-\langle x \rangle_{\!\mathrm{w(1)}}
}\right)}}
}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8325}%
$ \langle x\rangle^*$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8327}%
$\displaystyle {\sigma}_{\langle x \rangle_{\!\mathrm{w(1)}}}^{\mathrm{corrected}} = \mathsf{GoF}_{(1)}\ \sigma_{\langle x \rangle_{\!\mathrm{w(1)}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8331}%
$ O_j=C_j+\min{\ensuremath{\left({1,C_j}\right)}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8333}%
$ \sigma_j^2=C_j+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8335}%
$\displaystyle \langle x \rangle_{\!\mathrm{w(2)}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
{N_{\mathrm{obs}}^*}
}}}}{{\ensuremath{\displaystyle{
\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
}}}}{{\ensuremath{\displaystyle{
C_j+1
}}}}}}}
}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8337}%
$\displaystyle \sigma_{\langle x \rangle_{\!\mathrm{w(2)}}} = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
1
}}}}{{\ensuremath{\displaystyle{\sqrt{
\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
}}}}{{\ensuremath{\displaystyle{
C_j+1
}}}}}}}
}}}}}}}}=\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x \rangle_{\!\mathrm{w(2)}}}}}}{{\ensuremath{\displaystyle{
N_{\mathrm{obs}}^*
}}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8339}%
$\displaystyle \mathsf{GoF}_{(2)}=
\sqrt{
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
\!\!\!\!C_j+N_{\mathrm{obs}}^*
-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
{\ensuremath{\left[{
N_{\mathrm{obs}}^*
}\right]}}^2
}}}}{{\ensuremath{\displaystyle{ \mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
}}}}{{\ensuremath{\displaystyle{
C_j+1
}}}}}}} }}}}}}}
}}}}{{\ensuremath{\displaystyle{
N_{\mathrm{obs}}^*-1
}}}}}}}
}
=\sqrt{
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*-1}}}}}}}
{\ensuremath{\left({
\langle x\rangle^*-\langle x \rangle_{\!\mathrm{w(2)}}+1
}\right)}}
}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8343}%
$\displaystyle {\sigma}_{\langle x \rangle_{\!\mathrm{w(2)}}}^{\mathrm{corrected}} = \mathsf{GoF}_{(2)}\ \sigma_{\langle x \rangle_{\!\mathrm{w(2)}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8351}%
$\displaystyle \epsilon_x = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_x}}}}{{\ensuremath{\displaystyle{x}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8355}%
$ O(\epsilon_x^2)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8359}%
$ \propto\epsilon_x$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8363}%
$ O(\epsilon_x^3)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8367}%
$ N_{\mathrm{obs}}=2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8369}%
$ C_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8371}%
$ C_2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8373}%
$\displaystyle \langle x \rangle={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{C_1+C_2}}}}{{\ensuremath{\displaystyle{2}}}}}}}; \qquad \sigma_x={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sqrt{C_1+C_2}}}}}{{\ensuremath{\displaystyle{2}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8375}%
$\displaystyle \langle x \rangle_{\mathrm{w(2)}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{2(C_1+1)(C_2+1)}}}}{{\ensuremath{\displaystyle{C_1+C_2+2}}}}}}}; \qquad
\sigma_{\langle x \rangle_{\mathrm{w(2)}}}=\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{(C_1+1)(C_2+1)}}}}{{\ensuremath{\displaystyle{C_1+C_2+2}}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8377}%
$ C_1,C_2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8379}%
$ \lambda$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8381}%
$\displaystyle E{\ensuremath{\left({\langle x \rangle}\right)}} = \mathop{\sum}_{m,n=0}^{+\infty}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{n+m}}}}{{\ensuremath{\displaystyle{2}}}}}}}P(n)P(m)=\mathop{\sum}_{m=0}^{+\infty}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{m}}}}{{\ensuremath{\displaystyle{2}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\lambda^m{\ensuremath{\mathrm{e}}}^{-\lambda}}}}}{{\ensuremath{\displaystyle{m!}}}}}}}
+\mathop{\sum}_{n=0}^{+\infty}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{n}}}}{{\ensuremath{\displaystyle{2}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\lambda^n{\ensuremath{\mathrm{e}}}^{-\lambda}}}}}{{\ensuremath{\displaystyle{n!}}}}}}}
=\lambda
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8383}%
$\displaystyle E{\ensuremath{\left({\sigma_x^2}\right)}} = \mathop{\sum}_{m,n=0}^{+\infty}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{n+m}}}}{{\ensuremath{\displaystyle{4}}}}}}}P(n)P(m)={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
\lambda}}}}{{\ensuremath{\displaystyle{2}}}}}}};\qquad E{\ensuremath{\left({\sigma_x}\right)}} =\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\lambda}}}}{{\ensuremath{\displaystyle{2}}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8385}%
$\displaystyle \langle x \rangle_{\mathrm{w(2)}}=\langle x \rangle + 1 -{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{(C_1-C_2)^2}}}}{{\ensuremath{\displaystyle{4(\langle x \rangle+1)}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8387}%
$\displaystyle E{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{(C_1-C_2)^2}}}}{{\ensuremath{\displaystyle{4(\langle x \rangle+1)}}}}}}}}\right)}} =
\mathop{\sum}_{m,n=0}^{+\infty}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{(n-m)^2}}}}{{\ensuremath{\displaystyle{2(n+m+2) }}}}}}}P(n)P(m)={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{e}}}^{-2\lambda}}}}}{{\ensuremath{\displaystyle{2}}}}}}}
\mathop{\sum}_{m,n=0}^{+\infty}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{(n-m)^2}}}}{{\ensuremath{\displaystyle{(n+m+2) }}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\lambda^{n+m}}}}}{{\ensuremath{\displaystyle{n!m!}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8389}%
$ s=n+m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8391}%
$ s=0\ldots +\infty$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8393}%
$ n-m=s-2k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8395}%
$ k=0\ldots s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8397}%
$\displaystyle E{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{(C_1-C_2)^2}}}}{{\ensuremath{\displaystyle{4(\langle x \rangle+1)}}}}}}}}\right)}} =
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{e}}}^{-2\lambda}}}}}{{\ensuremath{\displaystyle{2}}}}}}}
\mathop{\sum}_{s=0}^{+\infty}
\mathop{\sum}_{k=0}^{s}
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{(s-2k)^2(s+1)}}}}{{\ensuremath{\displaystyle{(s+2)! }}}}}}}{\lambda^{s}}
\binom{s}{k}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{2}}}}}}}-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{2\lambda}}}}}}}+{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1-{\ensuremath{\mathrm{e}}}^{-2\lambda}}}}}{{\ensuremath{\displaystyle{4\lambda^2}}}}}}}
%{n!m!}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8399}%
$\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{E{\ensuremath{\left({\langle x \rangle_{\mathrm{w(2)}}-\langle x \rangle}\right)}}}}}}{{\ensuremath{\displaystyle{E{\ensuremath{\left({\langle x \rangle}\right)}}}}}}}}}=
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{2\lambda}}}}}}}+{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{2\lambda^2}}}}}}}-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1-{\ensuremath{\mathrm{e}}}^{-2\lambda}}}}}{{\ensuremath{\displaystyle{4\lambda^3}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8401}%
$ \langle x \rangle$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8403}%
$\displaystyle \epsilon = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_x}}}}{{\ensuremath{\displaystyle{\langle x \rangle}}}}}}} =
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\lambda^{1/2}}}}}{{\ensuremath{\displaystyle{\sqrt{2} \lambda}}}}}}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{\sqrt{2\lambda}}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8405}%
$\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{E{\ensuremath{\left({\langle x \rangle_{\mathrm{w(2)}}-\langle x \rangle}\right)}}}}}}{{\ensuremath{\displaystyle{E{\ensuremath{\left({\langle x \rangle}\right)}}}}}}}}}=
O(\epsilon^2)
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8411}%
$ \lambda=1,10,100,\ldots,1000000$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8415}%
$ N=10^8$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8421}%
$ \xi_\lambda=\sqrt{\lambda/N}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8423}%
$ \epsilon_\lambda=\sqrt{\lambda/N}/\lambda=1/\sqrt{N\lambda}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8425}%
$ E_\lambda$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8429}%
$ e_\lambda=E_\lambda/\lambda$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8431}%
$ e_\lambda/\epsilon_\lambda$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8433}%
$ \lambda =$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8435}%
$ \xi_\lambda = $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8437}%
$ \epsilon_\lambda$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8439}%
$ {\langle x \rangle_{\!\mathrm{w(1)}}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8441}%
$ {\langle x \rangle_{\!\mathrm{w(2)}}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8449}%
$ e_\lambda$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8793}%
$ {\langle x \rangle_{\!\mathrm{w(1)}}}\ :$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8797}%
$ {\langle x \rangle^*}\ $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8799}%
$ \lambda<100$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8801}%
$ {\langle x \rangle}\ $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8827}%
$\displaystyle X_0=\eta_0 C_0
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8829}%
$ \eta_0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8831}%
$ X$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8833}%
$\displaystyle P'(X)=P(X/\eta_0)=P(n)\qquad\Biggl|\Biggr.\qquad \frac{X}{\eta_0}\equiv n\in\mathbb{Z}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8837}%
$\displaystyle \langle X\rangle=\mathop{\sum}_{n=0}^{+\infty}
\eta_0 nP(n)=\eta_0 C_0=X_0\ ;
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8839}%
$\displaystyle \langle X^2\rangle=\mathop{\sum}_{n=0}^{+\infty}
\eta_0^2 n^2 P(n)=\eta_0^2(C_0^2+C_0)=X_0^2+\eta_0 X_0\ ;
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8841}%
$\displaystyle \sigma_X=\sqrt{\langle X^2\rangle-\langle X\rangle^2}=\sqrt{\eta_0 X_0}=\eta_0\sqrt{C_0}=\sqrt{\eta_0}\sqrt{X_0}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8843}%
$ \sigma_X=\sqrt{\langle X\rangle}=\sqrt{X_0}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8845}%
$\displaystyle \sigma_X=\sqrt{\eta_0}\sqrt{X_0}=\eta_0\sqrt{C_0}=\eta_0\sigma_{C_0}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8849}%
$ \sigma_{\eta_0}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8851}%
$\displaystyle \widehat{P}(\eta)={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
{\ensuremath{\mathrm{e}}}^{
-\frac{1}{2}
{\ensuremath{\left({
\frac{\eta-\eta_0}{\sigma_{\eta_0}}
}\right)}}^2
}
}}}}{{\ensuremath{\displaystyle{
\sigma_{\eta_0}\sqrt{2\pi}
}}}}}}}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8853}%
$\displaystyle \int_{-\infty}^{+\infty}{\ensuremath{\mathrm{d}{\eta}\, }}\mathop{\sum}_{n=0}^{+\infty}
P(n)\widehat{P}(\eta)=1\ ;
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8855}%
$\displaystyle \langle X\rangle=\int_{-\infty}^{+\infty}{\ensuremath{\mathrm{d}{\eta}\, }}\mathop{\sum}_{n=0}^{+\infty}
\widehat{P}(\eta)\eta nP(n)=
\mathop{\sum}_{n=0}^{+\infty}
nP(n)
\int_{-\infty}^{+\infty}{\ensuremath{\mathrm{d}{\eta}\, }} \widehat{P}(\eta)\eta
=
\eta_0 C_0=X_0\ ;
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8857}%
$\displaystyle \langle X^2\rangle=\int_{-\infty}^{+\infty}{\ensuremath{\mathrm{d}{\eta}\, }}\mathop{\sum}_{n=0}^{+\infty}
\widehat{P}(\eta)\eta^2 n^2 P(n)=
\int_{-\infty}^{+\infty}{\ensuremath{\mathrm{d}{\eta}\, }}\widehat{P}(\eta)\eta^2
\mathop{\sum}_{n=0}^{+\infty}
n^2 P(n)
=
(\eta_0^2+\sigma_{\eta_0}^2)(C_0^2+C_0)\ ;
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8859}%
$\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_X}}}}{{\ensuremath{\displaystyle{\langle X\rangle}}}}}}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sqrt{\langle X^2\rangle-\langle X\rangle^2}}}}}{{\ensuremath{\displaystyle{\langle X\rangle}}}}}}}
={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sqrt{
\eta_0^2 C_0+\sigma_{\eta_0}^2C_0^2+\sigma_{\eta_0}^2 C_0
}}}}}{{\ensuremath{\displaystyle{\eta_0C_0}}}}}}}=
\sqrt{
{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ \sigma_{C_0}}}}}{{\ensuremath{\displaystyle{C_0}}}}}}}}\right)}}^2
+{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_{\eta_0}}}}}{{\ensuremath{\displaystyle{\eta_0}}}}}}}}\right)}}^2+{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_{\eta_0}}}}}{{\ensuremath{\displaystyle{\eta_0}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_{C_0}}}}}{{\ensuremath{\displaystyle{C_0}}}}}}}}\right)}}^2
}\approx\sqrt{
{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ \sigma_{C_0}}}}}{{\ensuremath{\displaystyle{C_0}}}}}}}}\right)}}^2
+{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_{\eta_0}}}}}{{\ensuremath{\displaystyle{\eta_0}}}}}}}}\right)}}^2
}
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{section}
\stepcounter{section}
\end{document}
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