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\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{tabular}{llr}
1 & {\bf Basic Definitions and Results} & 5 \\ 
1.1 & Basic Definitions and Results & 6 \\ 
1.2 & Bayes Theorem & 11 \\ 
1.3 & Continuous Distributions & 20 \\ 
1.4 & Functions of a Random Variable & 23 \\ 
1.5 & Properties of Joint Distribution Functions & 27 \\ 
1.6 & Means, Moments, Correlations & 29 \\ 
1.7 & Central Limit Theorem & 43 \\ 
&& \\ 
2 & {\bf Feefoefum in Las Vegas:} & 62 \\ 
& {\bf Combinatoric and Discrete Probability} & \\ 
2.1 & Binomial and Poisson Distributions & 63 \\ 
2.2 & Birth and Death Processes & 78 \\ 
&& \\ 
3 & {\bf Estimation of Parameters} & 89 \\ 
3.1 & Maximum Likelihood Principle & 90 \\ 
3.2 & $\chi^2$ Method & 121 \\ 
3.3 & Counter Physics & 132 \\ 
3.4 & Bubble Chamber & 139 \\ 
\end{tabular}
}

{\newpage
\clearpage
\samepage \begin{tabular}{llr}
4 & {\bf Minimization} & 157 \\ 
4.1 & Statement of Problem & 158 \\ 
4.2 & Derivative Formalism for $\chi^2$ & 159 \\ 
4.3 & The Derivative Method for $\chi^2$ & 165 \\ 
4.4 & The Real World and $\chi^2$ Minimization & 168 \\ 
&& \\ 
5 & {\bf Goodness of Fit} & 180 \\ 
5.1 & The $\chi^2$ Distribution & 181 \\ 
5.2 & Confidence Level & 185 \\ 
5.3 & Bartlett's $S$ Function & 189 \\ 
5.4 & Hypothesis Testing & 191 \\ 
5.5 & Student's $t$ Distribution & 194 \\ 
5.6 & Robust Estimation & 197 \\ 
&& \\ 
6 & {\bf Generation of Random Numbers} & 211 \\ 
\end{tabular}
}

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{\newpage
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\samepage \psfig{figure=fig1.phys.ps,clip=true,width=6.5in}
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{\newpage
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{\newpage
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\samepage \setbox\sizebox=\hbox{$P\left( S_1 + S_2 + \ldots \right) = P\left( S_1 \right) + P\left( S_2 
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{\newpage
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{\newpage
\clearpage
\samepage \begin{eqnarray*}S_1 + S_2 & = & \left( S_1 - S_1\,S_2 \right) + S_2 \\ 
P\left( S_1 + S_2 \right) & = & P\left( S_1 - S_1\,S_2 \right) + 
P\left( S_2 \right) 
\end{eqnarray*}
}

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{\newpage
\clearpage
\samepage \begin{displaymath}\lim_{N\rightarrow\infty} \frac{N_h}{N} = \frac{1}{2}
\end{displaymath}
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$N_h$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3436: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \begin{displaymath}P{\rm (heads)} = \lim_{N\rightarrow\infty} \frac{N_h}{N} = \frac{1}{2}
\end{displaymath}
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$P(S) = 1/2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3440: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \psfig{figure=fig2.phys.ps,clip=true,width=4.5in}
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{\newpage
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{\newpage
\clearpage
\samepage \begin{eqnarray*}N_1 & = & n(A-AB) \\ 
N_2 & = & n(B-AB) \\ 
N_3 & = & n(AB) \\ 
N_4 & = & n\left( (A+B)^* \ \mbox{complement of}\ (A+B)\right)
\end{eqnarray*}
}

{\newpage
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{\newpage
\clearpage
\samepage \begin{eqnarray}P(A) & = & \lim_{N\rightarrow\infty} \frac{N_1 + N_3}{N} \nonumber \\ 
P(B) & = & \lim_{N\rightarrow\infty} \frac{N_2 + N_3}{N} \nonumber \\ 
P(A+B) & = & \lim_{N\rightarrow\infty} \frac{N_1 + N_2 + N_3}{N} 
\end{eqnarray}
}

\stepcounter{subsection}
{\newpage
\clearpage
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{\newpage
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\samepage \setbox\sizebox=\hbox{$P(SA)/P(A)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3484: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \begin{eqnarray}P(A/B) & = & \frac{N_3}{N_2 + N_3} \hspace{1em} \mbox{as}\ N\rightarrow\infty 
\nonumber \\ 
P(B/A) & = & \frac{N_3}{N_1 + N_3} \hspace{1em} \mbox{as}\ N\rightarrow\infty 
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}P(B) P(A/B) = P(A) P(B/A)
\end{equation}
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\samepage \setbox\sizebox=\hbox{$P(C) = P(A) P(B)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3512: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{n(C)}{n_{\rm tot}} = \frac{n(A)}{n_{\rm tot}} \cdot 
\frac{n(B)}{n_{\rm tot}}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}n_{\rm tot} = \frac{n(A) n(B)}{n(C)}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(A/B) = \frac{P(AB)}{P(B)} = P(A)
\end{displaymath}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}B_1 = 0 & \hspace{1em} & \mbox{ill} \\ 
B_2 = 1 & \hspace{1em} & \mbox{well}
\end{eqnarray*}
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$B=B_1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3536: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$B_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3538: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(B/A) = \frac{P(B) P(A/B)}{P(A)}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$B=B_1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3540: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$B=B_1 + B_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3544: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$P(B/A) = P(B_1/A) + P(B_2/A)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3546: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}&&P\left(B_2\right)\ \ P\left(A/B_2\right)\ \ \ P\left(B_1\right)
\ P\left(A/B_1\right) \\ 
P(B/A) & = & \frac{\lbrack .999 \times 0.02\delta (1-x) + .001 \times 
.99\delta (x) \rbrack}{\lbrack P(A) = .001 \times .99 + .02 \times
.999 \rbrack} \\ 
&& \\ 
P\left(B_1/A\right) & = & \mbox{``ill''} = .047 \\ 
&& \\ 
P\left(B_2/A\right) & = & \mbox{``well''} = .953
\end{eqnarray*}
}

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{\newpage
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\samepage \setbox\sizebox=\hbox{% latex2html id marker 3558
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\samepage \setbox\sizebox=\hbox{$P(A/B)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3574: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \begin{displaymath}P(B/A) = P(A/B) \frac{P(B)}{P(A)}
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{\newpage
\clearpage
\samepage \begin{displaymath}\frac{P\left(B_1/A\right)}{P\left(B_2/A\right)} = \frac{P\left(A/B_1\right)
P\left(B_1\right)}{P\left(A/B_2\right) P\left(B_2\right)}
\end{displaymath}
}

{\newpage
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{\newpage
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\samepage \begin{displaymath}F(x) = \int\limits^x_{-\infty} p(x) dx
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}

{\newpage
\clearpage
\samepage \begin{displaymath}p(x) = \frac{1}{2} \left\lbrack \delta (x) + \delta (x-1) \right\rbrack
\end{displaymath}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}p(x) = \frac{1}{\sqrt{2\pi} \sigma} \exp \left\lbrack - \frac{1}{2}
\frac{(x-\mu)^2}{\sigma^2} \right\rbrack
\end{equation}
}

{\newpage
\clearpage
\samepage \psfig{figure=fig3.phys.ps,clip=true,width=3.0in}
}

\stepcounter{subsection}
{\newpage
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\samepage \begin{displaymath}\int\limits^{+\infty}_{-\infty} p(x) dx = 1
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}F(x) = \Phi (x) = \frac{1}{\sqrt{2\pi} \sigma} {\displaystyle 
\int\limits^x_{-\infty}} dy \exp \left\lbrack - 
\frac{(y-\mu)^2}{2\sigma^2}\right\rbrack
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}{\rm erf}(x) = \frac{2}{\sqrt{\pi}} \int\limits^x_0 \exp \left( -y^2 \right) dy
\end{displaymath}
}

\stepcounter{subsection}
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}

{\newpage
\clearpage
\samepage \begin{equation}p(x) = q\left( f(x)\right) | f^\prime (x) |
\end{equation}
}

{\newpage
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}

{\newpage
\clearpage
\samepage \begin{eqnarray*}x & \rightarrow & p(x) dx \\ 
y & \rightarrow & q(y) dy
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r(x,y)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3696: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\int\limits r(x,y)dy & = & p(x) \\ 
\int\limits r(x,y)dx & = & q(y)
\end{eqnarray*}
}

\stepcounter{subsection}
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}

{\newpage
\clearpage
\samepage \begin{displaymath}T(X/Y) = \frac{Pr (XY)}{Pr (Y)} = \frac{r(x,y)}{q(y)}\ \frac{dx\,dy}{dy}
\end{displaymath}
}

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\clearpage
\samepage \setbox\sizebox=\hbox{$t(x|y) = r(x,y)/q(y)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3724: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \begin{displaymath}s(y|x) = \frac{r(x,y)}{p(x)}
\end{displaymath}
}

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{\newpage
\clearpage
\samepage \begin{equation}t(x|y) q(y) = s(y|x)p(x)
\end{equation}
}

{\newpage
\clearpage
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{\newpage
\clearpage
\samepage \begin{displaymath}r(x,y) = \delta \left\lbrack y-f(x)\right\rbrack p(x).
\end{displaymath}
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{\newpage
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\end{equation}
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{\newpage
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\samepage \begin{equation}d(w) = \int\!\!\int dx\, dy\ \delta \left\lbrack w - h(x,y) 
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\samepage \setbox\sizebox=\hbox{$\left\langle (x-\langle x\rangle
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{\newpage
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\samepage \setbox\sizebox=\hbox{$\sigma_x$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3794: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V_x = \sigma^2_x$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3796: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma_x$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3798: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V_x$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3800: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}V_x = \left\langle x^2 \right\rangle - 2\langle x\rangle \langle x\rangle
+ \langle x\rangle^2 = \left\langle x^2\right\rangle - \langle x\rangle^2
\end{equation}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bullet$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3806: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$p(x_1, \ldots x_n)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3810: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\left\langle x_{k} \right\rangle = \int\limits \otimes\,
dx_i \hspace{2em} x_k\, p(x_1 \ldots x_n)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$n\times n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3814: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\mbox{\bf M}_{\,kl} = \int\limits \otimes\, dx_i \ \ \left\lbrack
\mbox{\bf x}_{\,k} - \left\langle \mbox{\bf x}_{\,k} \right\rangle
\right\rbrack \left\lbrack \mbox{\bf x}_{\,l} - \left\langle \mbox{\bf
x}_{\,l} \right\rangle \right\rbrack p\left(x_1 \ldots x_n \right)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}M_{kl} = \left\langle x_k x_l \right\rangle - \left\langle x_k \right\rangle
\left\langle x_l \right\rangle
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$y_k\, M_{kl}\, y_l$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3816: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{${\bf \rho\,}_{kl} = M_{kl}/\sqrt{M_{kk}\, M_{ll}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3826: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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}

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{\newpage
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{\newpage
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{\newpage
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}

{\newpage
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\samepage \setbox\sizebox=\hbox{$M_{klmn\ldots}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3840: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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{\newpage
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}

\stepcounter{subsection}
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{\newpage
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}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$= \sqrt{\mbox{diagonal
elements of moment matrix}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3860: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf $y$} = T\Underline{$x$}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3866: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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}

{\newpage
\clearpage
\samepage \begin{eqnarray}\left\langle y_k y_l\right\rangle & = & T_{kk^\prime} \left\langle x_{k^\prime}
x_{l^\prime} \right\rangle T^T_{l^\prime l} \hspace{2em} \mbox{or} \nonumber \\ 
M_y & = & TM_x T^T
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$M_y$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3874: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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{\newpage
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}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}x_1 & = & \rho_{00} \\ 
x_2 & = & \rho_{11} \\ 
x_3 & = & \rho_{10}
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3892: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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}

{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 456
W(\theta , \varphi ) & = & 1+\sum^3_{i=1} x_i X_i (\theta, \varphi ), 
\hspace{2em} \mbox{or} \nonumber \\ 
W(\theta , \varphi ) & = & 1+\sum^3_{i=1} y_i Y_i (\theta, \varphi )
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$y_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3896: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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}

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\samepage \setbox\sizebox=\hbox{% latex2html id marker 3900
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\samepage \setbox\sizebox=\hbox{% latex2html id marker 3902
$Y_i
(\theta , \varphi )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3902: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 462
X_1 = \cos\theta\phantom{^2} & & Y_1 = P_1 ( \cos\theta ) \\ 
X_2 = \cos^2\theta & & Y_2 = P_2 ( \cos\theta ) \\ 
X_3 = \cos^3\theta & & Y_3 = P_3 ( \cos\theta )
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P_l$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3904: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \psfig{figure=fig4.phys.ps,clip=true,width=5.5in}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}b & = & \frac{\left( x_2 - x_1 \right)}{\left( z_2 - z_1 \right)} \\ 
a & = & \frac{z_2 x_1 - z_1 x_2}{z_2 - z_1}
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3914: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\left\langle (b-\langle b\rangle ) (a - \langle a\rangle ) \right\rangle = 
- \frac{z_2 \sigma^2_1 - z_1 \sigma^2_2}{\left( z_1 - z_2 \right)^2}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3920: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma_1 = \sigma_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3926: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$z_1 = -z_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3928: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \psfig{figure=fig5.phys.ps,clip=true,width=4.0in}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}% latex2html id marker 3940
\frac{d\sigma}{dt\,dx} = g(t) (1-x)^{1-2\alpha(t)}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x\leq 1 (.5 \leq x \leq 1\ \mbox{in the experiment})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3942: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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}

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\samepage \setbox\sizebox=\hbox{% latex2html id marker 3948
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\stepcounter{subsection}
{\newpage
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}

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\samepage \setbox\sizebox=\hbox{% latex2html id marker 3952
$\alpha(t)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3952: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \psfig{figure=fig6.phys.ps,clip=true,width=4.5in}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3954
${\displaystyle \int\limits^1_{.5}} g(t)
(1-x)^{1-2\alpha(t)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3954: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}\sigma^2_{y_i} = \sum^n_{j=1} \left( T_{ij}\right)^2 \sigma^2_{x_j}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3964: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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}

\stepcounter{subsection}
{\newpage
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_y = a_1^2 \sigma_1^2 + a_2^2 
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{\newpage
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}

{\newpage
\clearpage
\samepage \begin{displaymath}\sigma^2_y \leq \left( | a_1 \sigma_1 | + | a_2 \sigma_2 | \right)^2
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\tilde\sigma_y = \sqrt{a_1^2 \sigma_1^2 + a_2^2 \sigma_2^2}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma_y$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3982: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}0\leq \sigma_y \leq \sqrt{2} \tilde\sigma_y
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$| a_1 \sigma_1 | = |a_2
\sigma_2|$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3984: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$|a_1 \sigma_1|/|a_2 \sigma_2| \neq 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3986: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$a_1 \sigma_1 = 0 \Rightarrow \sigma_y = \tilde\sigma_y$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3988: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}0\leq \sigma_y \leq \sqrt{n} \tilde\sigma_y
\end{displaymath}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}p(x) = \frac{1}{\sqrt{2\pi} \sigma} \exp \left\lbrace - \frac{[x-\mu ]^2}{2
\sigma^2}\right\rbrace
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\langle x\rangle & = & \int\limits^{+\infty}_{-\infty} x\, p(x) dx \\ 
& = & \mu + \int\limits^{+\infty}_{-\infty} (x-\mu) p(x) dx \\ 
& = & \mu
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3998: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}& = & \frac{1}{\sqrt{2\pi} \sigma} \int\limits^{+\infty}_{-\infty}
(x-\mu)^2 p(x) dx \\ 
& = & \frac{\sigma^3}{\sqrt{2\pi} \sigma} \int\limits^{+\infty}_{-\infty} dy\
\frac{y^2}{\sigma^3}\ \exp \left\lbrack\frac{-y^2}{2\sigma^2}\right\rbrack
\end{eqnarray*}
}

{\newpage
\clearpage
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{\newpage
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}

{\newpage
\clearpage
\samepage \begin{eqnarray*}Variance & = & \frac{\sigma^3}{\sqrt{2\pi} \sigma}\ 
\frac{\partial}{\partial\sigma} \left\lbrace \int\limits^{+\infty}_{-\infty} 
e^{-y^2/2\sigma^2} dy \right\rbrace \\ 
& = & \frac{\sigma^2}{\sqrt{2\pi}}\ \frac{\partial}{\partial\sigma} 
\left\lbrack \sqrt{2\pi} \sigma \right\rbrack \\ 
& = & \sigma^2
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4004: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
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{\newpage
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}

{\newpage
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}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4012: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\langle y\rangle$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4014: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4016: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\left\langle (y-\mu )^2 \right\rangle & = & \frac{1}{n^2} \sum_{i,j} 
\left\langle \left( x_i - \mu \right) \left( x_j - \mu \right) \right\rangle \\ 
& = & \frac{\sigma^2}{n}
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4020: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\sigma}{\sqrt{n}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4022: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$n\rightarrow\infty$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4024: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$1/n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4028: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4036: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\varphi (k) & = & \int\limits^{+\infty}_{-\infty} dx\, e^{ikx} p(x) \\ 
& = & 1 + ik\langle x\rangle - \frac{k^2}{2!} \left\langle x^2 \right\rangle
+ \ldots
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}p(x) = \frac{1}{2\pi} \int\limits^{+\infty}_{-\infty} dx\, e^{-ikx} \varphi (k)
\end{displaymath}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}w(h) = \int\limits^{+\infty}_{-\infty} \int\limits^{+\infty}_{-\infty} 
\delta (h-x-y)\, p(x)\, q(y)\ dx\, dy
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\varphi_h (k) & = & \int\limits^{+\infty}_{-\infty} dh\, w(h) e^{ikh} \\ 
& = & \int\limits^{+\infty}_{-\infty} \int\limits^{+\infty}_{-\infty} 
e^{ik(x+y)}\, p(x)\, q(y)\ dx\, dy \\ 
& = & \varphi_x (k) \varphi_y (k)
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\varphi (k)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4058: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\varphi (k) = \sum_n k^n E \frac{(ix)^n}{n!}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\varphi (k) = 1 + i\langle x\rangle k - \frac{\langle x^2 \rangle k^2}{2}
, \mbox{etc.}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\varphi (k) & = & \frac{1}{\sqrt{2\pi}\sigma} \int\limits^{+\infty}_{-\infty} 
\exp (ixk) \exp \left\lbrack - \frac{1}{2} 
\frac{(x-\mu)^2}{\sigma^2}\right\rbrack \\ 
& = & \exp \left\lbrace i\mu k - \frac{1}{2} k^2 \sigma^2 \right\rbrace
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}K(k) = \ln \varphi (k)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}K(k) = \langle x\rangle ik - \underbrace{\left( \left\langle x^2 \right\rangle
- \langle x\rangle^2 \right)}_{\sigma^2} \frac{k^2}{2}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x\rightarrow x+\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4062: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$ik\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4066: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4068: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$y = \frac{1}{n} {\displaystyle \sum^n_{i=1}} x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4070: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4072: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bullet$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4074: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4076: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4082: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bullet$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4084: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4086: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bullet$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4092: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4094: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_1 \ldots x_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4096: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4100: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma/\sqrt{n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4102: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4104: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4106: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4108: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4110: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4112: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_1 \ldots x_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4114: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\mu_i & = & \left\langle x_i \right\rangle \\ 
\sigma^2_i & = & \left\langle \left( x_i - \mu_i \right)^2 \right\rangle \\ 
\rho^3_i & = & \left\langle | x_i - \mu_i |^3 \right\rangle
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4116: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4118: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\rho^3 & = & \sum_{i=1}^n \rho_i^3 \hspace{2em} \mbox{and} \\ 
\Sigma^2 & = & \sum_{i=1}^n \sigma_i^2
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lim_{n\rightarrow\infty} \rho/\Sigma
\rightarrow 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4120: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bullet$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4124: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu = \sum^n_{i=1} \mu_i/n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4126: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bullet$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4128: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Sigma/n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4130: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lim_{n\rightarrow\infty} \rho/\Sigma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4132: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rightarrow 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4134: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4136: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho\propto
n^{1/3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4138: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\displaystyle \sum}\propto n^{1/2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4140: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}\varphi_y (k) = \left\lbrack \varphi_x \left(\frac{k}{n}\right) 
\right\rbrack^n
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\varphi_y$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4144: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\varphi_x$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4146: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\varphi_x (k) & = & \exp (ik\mu) \int\limits dx e^{ik(x-\mu)} p(x) \\ 
& = & \exp (ik\mu) \left\lbrack 1 - \frac{k^2 \sigma^2}{2} + O\left( k^3
\right) \right\rbrack
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}\varphi_y (k) & = & \exp (ik\mu) \left\lbrack 1 - \frac{k^2 \sigma^2}{2n^2} + 
O\left( \frac{k}{n} \right)^3 \right\rbrack^n \\ 
& = & e^{ik\mu} \left\lbrace \exp \left\lbrack - \frac{k^2 \sigma^2}{2n^2} + 
O\left( \frac{k}{n} \right)^3 \right\rbrack \right\rbrace^n \\ 
& = & \exp (ik\mu ) \exp \left\lbrace - \frac{k^2 \sigma^2}{2n} + 
O\left( \frac{k^3}{n^2} \right) \right\rbrace
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{k^3}{n^2} \rightarrow 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4156: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{k^2}{n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4158: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$n\rightarrow\infty$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4160: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}p(y) = \frac{1}{\sqrt{2\pi}}\ \frac{\sqrt{n}}{\sigma} \exp \left\lbrace -
\frac{n(x-\mu)^2}{2\sigma^2} \right\rbrace
\end{displaymath}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}I = \int\limits^b_a f(x) dx
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}I = (b-a) \int\limits^{+\infty}_{-\infty} f(x) p(x) dx
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}p(x) & = & \frac{1}{(b-a)} \hspace{2em} a\leq x\leq b \\ 
& = & 0 \hspace{4em} \mbox{outside}\ [a,b].
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}{\displaystyle \int} p(x)\, dx = 1
\end{displaymath}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\overline{I} = \frac{1}{n} {\displaystyle \sum^n_{i=1}}
f\left( x_i \right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4170: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4172: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4174: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$z_i = f\left( x_i \right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4176: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$z_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4178: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bullet$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4180: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu = {\displaystyle \int\limits^b_a} f(x) p(x)
dx$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4182: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bullet$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4184: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2 = {\displaystyle \int\limits^b_a}
f^2 (x) p(x) dx - \mu^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4186: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bullet$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4188: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\overline{I} \rightarrow \frac{I}{(b-a)} \pm
\frac{\sigma}{\sqrt{n}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4190: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(b-a)\overline{I}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4194: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(b-a)
\frac{\sigma}{\sqrt{n}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4196: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4202: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4206: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\int\limits^b_a f^2 (x) p(x) dx = \left\langle f^2 (x) \right\rangle
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{1}{n} \sum^n_{i=1} f^2 \left( x_i \right)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\displaystyle \sum} f^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4208: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\displaystyle \sum} f$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4210: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4212: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4214: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$g(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4216: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\mu_f & = & \langle f\rangle\,, \hspace{2em} \mu_g  =  \langle g\rangle \\ 
\sigma^2_f & = & \left\langle \left( f - \mu_f \right)^2 \right\rangle \\ 
\sigma_{fg} & = & \left\langle \left( f - \mu_f \right) \left( g - \mu_g
\right) \right\rangle \\ 
\sigma^2_g & = & \left\langle \left( g - \mu_g \right)^2 \right\rangle \\ 
M & = & \left\langle
\begin{array}{ll}
\sigma^2_f & \sigma_{fg} \\ 
\sigma_{fg} & \sigma^2_g
\end{array}
\right\rangle \hspace{2em} \mbox{moment matrix}
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}F & = & \frac{1}{n} \sum^n_{i=1} f\left( x_i \right) \\ 
G & = & \frac{1}{n} \sum^n_{i=1} g\left( x_i \right) 
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}p(F,G) \propto \exp \left\lbrace - \frac{n}{2} (y-\mu)^T B(y-\mu)
\right\rbrace
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$ y = \left\lbrack
\begin{array}{l}
F \\  G
\end{array}
\right\rbrack\,, \hspace{1em} \mu = \left\lbrack
\begin{array}{l}
\mu_f \\  \mu_g
\end{array}
\right\rbrack$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4222: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$B^{-1} = M$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4224: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\varphi \left( k_1,
k_2 \right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4226: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$F/G$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4230: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu_f/\mu_g$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4232: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{\langle f\rangle}{\langle g\rangle} \sqrt{\frac{1}{n}
\left\langle \left( \frac{f}{\langle f\rangle} - \frac{g}{\langle
g\rangle} \right)^2 \right\rangle}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bullet$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4234: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$F/G$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4238: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$=\mu_f$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4240: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma =
\sqrt{\frac{1}{n} \sigma^2_f}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4242: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
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}

{\newpage
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}

{\newpage
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}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}\begin{array}{rl}
f = 1 & \hspace{1em} \mbox{event passes selection criteria} \\ 
= 0 & \hspace{1em} \mbox{event fails criterion.} \\ 
\end{array}
\end{displaymath}
}

{\newpage
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\samepage \setbox\sizebox=\hbox{% latex2html id marker 4258
$f=\theta g$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4258: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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{\newpage
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}

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$\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4268: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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$\theta=1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4272: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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$\theta=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4276: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{% latex2html id marker 4278
$\langle\theta\rangle=p$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4278: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\langle f\rangle = p\langle
g\rangle$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4280: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$N_{\rm hist}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4286: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 854
\frac{F}{G} & = & \frac{{\displaystyle \sum^N_{i=1}}\, \theta_i\,
g_i}{\displaystyle \sum^N_{i=1}\, g_i} \\ 
& = & \frac{\mbox{Number of Events in Bin}}{\mbox{Number of Events in
Histogram}}
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{\langle f\rangle}{\langle g\rangle}\ \sqrt{\frac{1}{N} \left\langle 
\left( \frac{f}{\langle f\rangle} - \frac{g}{\langle g\rangle} \right)^2
\right\rangle}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 870
& = & p\ \sqrt{\frac{1}{N\, \langle f\rangle^2}\ \left\langle 
(g\theta - g\,p)^2 \right\rangle} \\ 
& = & p\ \sqrt{\frac{\langle g^2 \rangle}{N\langle f\rangle^2}
\left\langle (\theta - p)^2 \right\rangle}
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 4288
$\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4288: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{% latex2html id marker 4290
$\langle\theta^2 \rangle =
\langle\theta\rangle = p$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4290: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{% latex2html id marker 4292
$\langle (\theta - p)^2 \rangle = p(1-p)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4292: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
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\samepage \setbox\sizebox=\hbox{$\langle g^2 \rangle = \langle g\rangle$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4294: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$\sqrt{p(1-p)}/\sqrt{N\langle
g\rangle}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4296: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$\sqrt{N\langle g\rangle} = N_{\rm hist} - $}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4298: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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$\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4300: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$N_{\rm hist}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4302: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$\pm \sqrt{N_{\rm hist}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4304: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$N_{\rm
hist}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4308: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 894
\langle\theta\rangle & = & \frac{1}{N_{\rm hist}} \sum^{N_{\rm
hist}}_{i=1} \theta \left( x_i \right) \\ 
\left\langle \theta^2 \right\rangle & = & \frac{1}{N_{\rm hist}}
\sum^{N_{\rm hist}}_{i=1} \theta^2 \left( x_i \right) =
\langle\theta\rangle \\ 
\sigma & = & \left\langle \theta^2 \right\rangle - \langle
\theta\rangle^2 = p - p^2
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma/\sqrt{N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4310: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

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}

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$\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4314: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
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$\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4318: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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$p(\theta )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4320: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 4322
$p(\theta) = {\displaystyle
\sum^\infty_{n=0}} a_n P_n (\cos\theta)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4322: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{% latex2html id marker 4324
$P_n (\cos\theta)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4324: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 915
\left\langle P_{n_1} (\cos\theta)\right\rangle & = & \int\limits P_{n_1}
(\cos\theta) p(\theta) d\theta \\ 
& = & a_{n_1} \frac{2}{2n+1}
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$a_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4326: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 4328
$\left\langle P_{n_1}
(\cos\theta)\right\rangle$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4328: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

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\samepage \setbox\sizebox=\hbox{% latex2html id marker 4330
$\theta_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4330: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}% latex2html id marker 4316
\left\langle P_n (\cos\theta)\right\rangle = \frac{1}{N} \sum^N_{k=1}
P_n \left( \cos\theta_k \right)
\end{displaymath}
}

\stepcounter{subsection}
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\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 4332
$P_{n_1} (\cos\theta)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4332: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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\samepage \setbox\sizebox=\hbox{% latex2html id marker 4334
$P_{n_2} (\cos\theta)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4334: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

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\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 4336
$\left\langle P_{n_1}
(\cos\theta)\right\rangle$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4336: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

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$\left\langle P_{n_2}
(\cos\theta)\right\rangle$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4338: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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$\left\lbrace\theta_k \right\rbrace$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4340: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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\samepage \setbox\sizebox=\hbox{% latex2html id marker 4342
$\theta_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4342: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 934
f & = & P_{n_1} (\cos\theta) \\ 
g & = & P_{n_2} (\cos\theta) \\ 
F & = & \frac{1}{N} \sum_k P_{n_1} \left(\cos\theta_k \right) \\ 
G & = & \frac{1}{N} \sum_k P_{n_2} \left(\cos\theta_k \right) 
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 4348
${\displaystyle \int\limits}
p(\theta) P_{n_1} (\cos\theta) P_{n_2} (\cos\theta) = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4348: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$a_{n_1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4350: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$a_{n_2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4352: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$w_i (i=1 \ldots N)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4358: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{{\displaystyle 
\int\limits} x\, w(x) dx}{{\displaystyle \int\limits} w(x) dx}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4364: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\langle x\rangle = \frac{{\displaystyle \sum^N_{i=1}} w_i\, x_i = 
S_3}{{\displaystyle \sum^N_{i=1}} w_i = S_1}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\sigma & = & \frac{S_3}{S_1} \sqrt{\frac{1}{N} \left\langle \left(
\frac{Nwx}{S_3} - \frac{Nw}{S_1} \right)^2 \right\rangle} \\ 
\sigma^2 & = & \frac{S^2_3}{S^2_1}\, N \left\langle \left\lbrack
\frac{w^2 x^2}{S^2_3} - \frac{2w^2 x}{S_3 S_1} + \frac{w^2}{S^2_1}
\right\rbrack \right\rangle \\ 
& = & \frac{1}{S^2_1} \left\lbrack S_6 - 2\langle x\rangle S_4 +
\langle x\rangle^2 S_2 \right\rbrack \\ 
S_2 & = & \sum^N_{i=1} w_i^2 \\ 
S_4 & = & \sum^N_{i=1} w_i^2\, x_i \\ 
S_6 & = & \sum^N_{i=1} w_i^2\, x_i^2
\end{eqnarray*}
}

\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bullet$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4370: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bullet$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4374: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$b(k,n,p)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4380: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
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\samepage \setbox\sizebox=\hbox{$b(k,n,p)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4390: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\underbrace{ht\, h\, tt \ldots}_n
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\left( \begin{array}{l} n \\  k \end{array} \right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4392: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p^k
q^{n-k}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4398: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}b(k,n,p) = \left( \begin{array}{l} n \\  k \end{array} \right) p^k q^{n-k}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$b(k,n,p)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4400: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}1 = (p+q)^n = \sum^n_{k=0} b(k,n,p).
\end{displaymath}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p=1/20$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4402: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q=19/20$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4404: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Pr\{ \mbox{Frank Nagy smiles once and once only} \}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4408: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}& = & \left( \begin{array}{c} 1000 \\  1 \end{array} \right) \cdot
\frac{1}{20} \cdot \left( \frac{19}{20} \right) ^{999} \\ 
& = & 50\, e^{999\log_e 19/20} \hspace{1em} \mbox{which isn't very big} \\ 
& \approx & 50\, e^{-51} \\ 
& \approx & 10^{-19}
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$b(k,n,p)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4412: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$n\rightarrow\infty$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4414: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p\rightarrow 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4416: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$np=\lambda$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4418: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}b\left( k, n, \frac{\lambda}{n} \right) = \frac{n!}{k! (n-k)!} \left(
\frac{\lambda}{n} \right)^k \left( 1 - \frac{\lambda}{n} \right)^n
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{n!}{(n-k)!} = n(n-1) \ldots (n-k+1) \rightarrow n^k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4422: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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\samepage \setbox\sizebox=\hbox{$n\rightarrow \infty$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4424: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\left( 1 - \frac{\lambda}{n} \right)^{n-k} & \approx & \left( 1 -
\frac{\lambda}{n}\right)^n \rightarrow e^{-\lambda}. \hspace{1em} \mbox{So,}
\nonumber \\ 
b \left( k, n, \frac{\lambda}{n} \right) & \rightarrow & \frac{n^k}{k!}\,
\frac{\lambda^k}{n^k}\, e^{-\lambda} \nonumber \\ 
& = & P(k,\lambda) = e^{-\lambda} \frac{\lambda^k}{k!}
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(k,\lambda)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4428: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\displaystyle \sum^\infty_{k=0}} P(k,\lambda)=1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4432: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p=\frac{1}{20.50} =
\frac{1}{1000}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4436: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$np=\lambda=1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4442: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$e^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4444: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\left( \begin{array}{c} 1000 \\  1 \end{array} \right) \frac{1}{1000}
\left( \frac{999}{1000} \right)^{999} =
\left\lbrack \frac{999}{1000} \right\rbrack^{999}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\approx e^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4446: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$y={\displaystyle \sum^N_{i=1}} y_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4448: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$y_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4450: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4452: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lambda$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline4456: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
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{\newpage
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\samepage \setbox\sizebox=\hbox{${\displaystyle
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{\newpage
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}

{\newpage
\clearpage
\samepage \begin{eqnarray*}Pr & ( & y = \mbox{integer}\ r ) \\ 
& = & {\displaystyle \sum^\infty_{N=0}} \frac{\lambda^N 
e^{-\lambda}}{\lambda!} \cdot \frac{(N\mu)^r e^{-N\mu}}{r!}
\end{eqnarray*}
}

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\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}\mbox{Mean}\ = \lambda && \sigma^2 = \lambda \\ 
f_2 (x) = \sigma^2_x & - & \langle x \rangle = 0
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\langle y\rangle & = & 4\lambda \\ 
\left\langle y^2 \right\rangle & = & \left\langle (4x)^2 \right\rangle = 16
\left\langle x^2 \right\rangle \\ 
& = & 16 \left( \left\langle x^2 \right\rangle - \langle x\rangle^2 +
\langle x\rangle \right) \\ 
& = & 16 \left( \sigma_x^2 + \langle x\rangle^2 \right) \\ 
& = & 16 \left( \lambda + \lambda^2 \right) \\ 
f_2 (y) & = & \left\langle y^2 \right\rangle - \langle y\rangle^2 -
\langle y\rangle \\ 
& = & 16\left( \lambda + \lambda^2 \right) - 16\lambda^2 - 4\lambda \\ 
& = & 12\lambda > 0
\end{eqnarray*}
}

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{\newpage
\clearpage
\samepage \begin{eqnarray*}\mu & = & 0 \cdot (1-p) + 1\cdot p = p \\ 
\sigma^2 & = & \left\langle x^2 \right\rangle - \langle x\rangle^2 \\ 
& = & 0 \cdot (1-p) + 1\cdot p - p^2 = p(1-p)
\end{eqnarray*}
}

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}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\mu & = & \sum \exp(-\lambda) k \frac{\lambda^k}{k!} \\ 
& = & \lambda e^{-\lambda} \frac{\partial}{\partial\lambda} \left(
\frac{\sum \lambda^k}{k!} \right) \\ 
& = & \lambda e^{-\lambda} \frac{\partial}{\partial\lambda} \left(
e^\lambda \right) \\ 
& = & \lambda
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}\sigma^2 & = & \left\langle x^2 \right\rangle - \lambda^2 \\ 
& = & e^{-\lambda} \left( \frac{\lambda\partial}{\partial\lambda}
\right)^2 \left( e^\lambda \right) - \lambda^2 \\ 
& = & e^{-\lambda} \left( \frac{\lambda\partial}{\partial\lambda}
\right) \left(\lambda e^\lambda \right) - \lambda^2 \\ 
& = & \lambda + \lambda^2 - \lambda^2 = \lambda = \mu
\end{eqnarray*}
}

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{\newpage
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\end{displaymath}
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{\newpage
\clearpage
\samepage \begin{displaymath}\frac{dP_k}{dt} (t) = -\lambda P_k(t) + \lambda P_{k-1} (t)
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{\newpage
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{\newpage
\clearpage
\samepage \begin{tabular}{cclcr}
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$\bullet$ & & $E_n \rightarrow E_{n-1}$ & chance & $\mu_n dt + o(dt)$ \\ 
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{\newpage
\clearpage
\samepage \begin{eqnarray*}\frac{dP_n (t)}{dt} = & - & \left(\lambda_n + \mu_n \right) P_n (t) \\ 
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\end{eqnarray*}
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{\newpage
\clearpage
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\samepage \begin{displaymath}% latex2html id marker 4860
p_i \left( x_i \right) = c \exp \left\lbrace - \frac{1}{2\sigma^2_0}
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\stepcounter{subsection}
{\newpage
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\samepage \begin{eqnarray*}% latex2html id marker 1318
L\left( x_1 \ldots x_n \right) & = & \prod^n_{i=1} c \exp \left\lbrace - 
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& = & \tilde{c} \exp \left\lbrace - \frac{1}{2\sigma^2_0} \sum_i
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\samepage \begin{displaymath}% latex2html id marker 4882
L \left( x_i \ldots x_n \right) = Pr \left\lbrace x_1 \ldots x_n \vert
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P \left( \alpha \vert x_i , T \right) = P \left( x_i \vert \alpha , 
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\phi (\beta \vert T) d\beta & = & P(\alpha \vert T ) d\alpha ,
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L\left( x_1 \ldots x_n \right) = c \exp \left\lbrace -
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L(\alpha) & = & c\, \exp \left\lbrace - \frac{n}{2\sigma^2} \left(
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& - & \frac{1}{2\sigma^2} \left\lbrack \sum_i x_i^2 - \frac{1}{n}
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L(\alpha ) = \exp \left\lbrace \mbox{const.} -
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p (\varphi , \alpha ) = 
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&& maximizes $L$ \\ 
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&& maximizes $L$
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L(\alpha) = \frac{(1 + \alpha (\cos \varphi_1 + \cos \varphi_2 ) +
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\samepage \begin{eqnarray*}L(\tau) & \propto & \frac{1}{\tau}\, e^{-\frac{t_1}{\tau}}\ \ldots\ 
\frac{1}{\tau}\, e^{-\frac{t_n}{\tau}} , \hspace{1em} \mbox{or} \\ 
L(\tau) & = & \exp \left\lbrack - \frac{1}{\tau}\, {\displaystyle \sum^n_{i=1}}
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\end{eqnarray*}
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\clearpage
\samepage \begin{eqnarray*}\frac{dL}{d\tau} & = & 0 \hspace{1em} \mbox{at}\ \tau=\tau^* \ \
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\frac{1}{\tau^{*^2}}\, {\displaystyle \sum^n_{i=1}} t_i - \frac{n}{\tau^*} 
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\samepage \begin{eqnarray}\sigma^2 & = & \frac{1}{n} \left\lbrace \left\langle t^2_i
\right\rangle - \left\langle t_i \right\rangle^2 \right\rbrace
\nonumber \\ 
& = & \frac{1}{n} \left\lbrace \frac{1}{\tau_0} {\displaystyle \int} t^2
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\sigma & = & \frac{\tau_0}{\sqrt{n}}
\end{eqnarray}
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\samepage \begin{displaymath}L(\tau) \propto \exp \left\lbrace - \frac{1}{2\tilde{\sigma}^2}
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\end{displaymath}
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}

{\newpage
\clearpage
\samepage \begin{eqnarray}\frac{1}{\tilde{\sigma}^2} & = & - \frac{d}{d\tau^2} \ln L(\tau)
\big\vert_{\tau=\tau^*} \nonumber \\ 
& = & \frac{2}{\tau^3} \sum^n_{i=1} t_i - \frac{n}{\tau^2}
\big\vert_{\tau=\tau^*} \hspace{1em} \mbox{or} \nonumber \\ 
\tilde\sigma & = & \frac{\tau^*}{\sqrt{n}} 
\end{eqnarray}
}

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\samepage \begin{displaymath}% latex2html id marker 5252
\langle\alpha^* \rangle = \alpha_0 + O \left( \frac{1}{n} \right)
\end{displaymath}
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{\newpage
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\samepage \begin{displaymath}% latex2html id marker 5288
\ln L(\alpha) = \sum^n_{i=1} \ln P(x_i , \alpha)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}% latex2html id marker 5289
\ln L(\alpha) = L_0 + (\alpha - \alpha_0 ) L_1 + \frac{(\alpha -
\alpha_0)^2}{2} L_2 + \frac{(\alpha - \alpha_0)^3}{6} L_3 + \ldots
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 1649
L_0 & = & \sum^n_{i=1} \ln P(x_i , \alpha_0 ) \\ 
L_1 & = & \sum^n_{i=1} \frac{\partial}{\partial\alpha} \ln P(x_i ,
\alpha_0 ) \\ 
L_2 & = & \sum^n_{i=1} \frac{\partial^2}{\partial\alpha^2} \ln P(x_i ,
\alpha_0 ) , \hspace{1em} \mbox{etc.}
\end{eqnarray*}
}

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\samepage \begin{eqnarray}% latex2html id marker 1661
L_1 & + & (\alpha^* - \alpha_0 ) L_2 = 0 \hspace{1em} \mbox{i.e.,}
\nonumber \\ 
&& \alpha^* = \alpha_0 - \frac{L_1}{L_2}
\end{eqnarray}
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\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 1676
\langle \alpha^* \rangle & = & \alpha_0 - \langle L_1/L_2 \rangle \\ 
& = & \alpha_0 - \frac{\left\langle \frac{\partial}{\partial\alpha}
\ln P(x , \alpha_0 ) \right\rangle}{\left\langle 
\frac{\partial^2}{\partial\alpha^2} \ln P(x , \alpha_0 ) \right\rangle} 
\end{eqnarray*}
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\samepage \begin{displaymath}% latex2html id marker 5312
\langle\alpha^* \rangle = \alpha_0 - \frac{\left\langle
\frac{\partial}{\partial\alpha} \ln P(x , \alpha_0 )
\right\rangle}{\left\langle \frac{\partial^2}{\partial\alpha^2} \ln P(x , 
\alpha_0 ) \right\rangle}
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\samepage \begin{eqnarray*}% latex2html id marker 1693
\int\limits dx P(x, \alpha_0 ) \frac{\partial}{\partial\alpha} \ln P(x,
\alpha_0) & = & \int\limits dx \frac{\partial}{\partial\alpha} P(x, \alpha)
\big\vert_{\alpha=\alpha_0} \\ 
& = & \frac{\partial}{\partial\alpha} \left\lbrack \int\limits dx P(x,
\alpha) \right\rbrack \big\vert_{\alpha=\alpha_0} \\ 
& = & \frac{\partial}{\partial\alpha} \cdot 1 \\ 
& = & 0
\end{eqnarray*}
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\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 1713
\sigma^2_{\rm physics} & = & \frac{-1}{{\displaystyle \sum^n_{i=1}} \left\lbrack
\frac{\partial^2}{\partial\alpha^2} \ln P(x_i, \alpha_0) \right\rbrack} \\ 
\langle\sigma^2_{\rm physics}\rangle & = & \frac{-1}{n \left\langle
\frac{\partial^2}{\partial\alpha^2} \ln P(x_i, \alpha_0) \right\rangle}
\hspace{2em} \mbox{where} \\ 
\left\langle \frac{\partial^2}{\partial\alpha^2} \ln P(x,\alpha)
\right\rangle & = & \int\limits dx P \left\lbrace \frac{- (\partial P/ 
\partial\alpha)^2}{P^2} + \frac{\partial^2 P/\partial\alpha^2}{P}
\right\rbrace
\end{eqnarray*}
}

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\samepage \setbox\sizebox=\hbox{% latex2html id marker 5340
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{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 1735
\left\langle \sigma^2_{\rm physics} \right\rangle =
\frac{1}{\left\lbrace n {\displaystyle \int} dx \left( \frac{\partial
P}{\partial\alpha} \right)^2 / P \right\rbrace}
\end{equation}
}

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{\newpage
\clearpage
\samepage \begin{displaymath}\sigma^2 = \frac{1}{n} \left\langle\left( \frac{f}{\langle g\rangle}
- \frac{g\langle f\rangle}{\langle g\rangle^2} \right)^2\right\rangle
\end{displaymath}
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\samepage \begin{displaymath}% latex2html id marker 5343
\left\langle f^2 \right\rangle = {\displaystyle\int\limits} \frac{\left( 
\frac{\partial P}{\partial\alpha} \right)^2}{P^2} \cdot P dx
\end{displaymath}
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{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 1788
L_1 & = & \prod^{n_1}_{i=1} p (x_i, \alpha)\ \ \mbox{is likelihood of
experiment \# 1} \\ 
L_2 & = & \prod^{n_2}_{i=1} q (y_i, \alpha)\ \ \mbox{is likelihood of
experiment \# 2}
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}L = L_1 L_2
\end{displaymath}
}

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{\newpage
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\samepage \begin{displaymath}p_i = \frac{1}{\sqrt{2\pi} \sigma_i} \exp \left\lbrace \frac{- (x_i -
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\end{displaymath}
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{\newpage
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\samepage \begin{displaymath}L = \prod^N_{i=1} p_i \propto \exp \left\lbrace - \sum^N_{i=1}
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{\newpage
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\samepage \begin{equation}\chi^2 = \sum^N_{i=1} \frac{(x_i - \xi_i)^2}{\sigma^2_i}
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{\newpage
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{\newpage
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\samepage \begin{eqnarray*}x_1 & = & \frac{1}{n} \sum^m_{k=1} y_k \\ 
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\samepage \psfig{figure=fig11.phys.ps,clip=true,width=5.5in}
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{\newpage
\clearpage
\samepage \begin{eqnarray*}x_i & = & \mbox{const.}\ \ N_i \\ 
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{\newpage
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\samepage \begin{eqnarray*}\sum^N_{i=1} \frac{(x_i - \xi^*_i)}{\sigma^2_i}\ 
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{\newpage
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\clearpage
\samepage \begin{displaymath}b_m = \sum_i \frac{x_i C_{im}}{\sigma^2_i} = \sum_i
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\samepage \psfig{figure=fig12.phys.ps,clip=true,width=5.5in}
}

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{\newpage
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\samepage \begin{equation}p_i = e^{-\lambda_i t} \frac{(\lambda_i t)^{n_i}}{n_i!}
\end{equation}
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{\newpage
\clearpage
\samepage \begin{eqnarray}L & = & \prod^M_{i=1} p_i \hspace{2em} \mbox{or} \nonumber \\ 
&& \nonumber \\ 
\ln L & = & - \sum^M_{i=1} \ln n_i! - t \sum^M_{i=1} \lambda_i +
\sum^M_{i=1} n_i \ln (\lambda_i t)
\end{eqnarray}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}\frac{\partial (\ln L)}{\partial t} & = & 0 \hspace{2em} \mbox{or} \\ 
t & = & \frac{{\displaystyle\sum^M_{i=1}} n_i}{{\displaystyle\sum^M_{i=1}}
\lambda_i} \\ 
&& \\ 
& = & \frac{N}{\Lambda}
\end{eqnarray*}
}

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{\newpage
\clearpage
\samepage \begin{eqnarray*}\ln L & = & - \sum^M_{i=1} \ln n_i! - N + N \ln N \\ 
& + & \sum^M_{i=1} n_i \ln \left( \frac{\lambda_i}{\Lambda} \right)
\end{eqnarray*}
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{\newpage
\clearpage
\samepage \begin{eqnarray}\ln L & = & \sum^M_{i=1} n_i \ln \left( \frac{\lambda_i}{\Lambda}
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& = & \sum^N_{j=1} \ln \tilde{p} (C_j)
\end{eqnarray}
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{\newpage
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\end{displaymath}
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{\newpage
\clearpage
\samepage \begin{displaymath}\ln L = \mbox{const.}\ + \underbrace{\lbrace N \ln (t\Lambda) -
t\Lambda \rbrace}_{(a)} + \underbrace{\sum^M_{i=1} n_i \ln
(\lambda/\Lambda)}_{(b)}
\end{displaymath}
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{\newpage
\clearpage
\samepage \begin{eqnarray*}L & = & \prod_{\rm successes} p \mbox{(success)} \prod_{\rm failures}
p \mbox{(failure)} \\ 
& = & \prod^N_{i=1} \exp (-p_i\, t\, dx_i) (p_i\, dx_i\, t) \\ 
& \otimes & \prod^{M-N}_{j=1} \exp (-p_j\, dx_j\, t)
\end{eqnarray*}
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\end{displaymath}
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L = \exp \left\lbrack - \int\limits p(x)\, dx\, t \right\rbrack \prod^N_{i=1} p
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{\newpage
\clearpage
\samepage \begin{eqnarray*}\frac{\partial L}{\partial t} & = & 0 \hspace{2em} \mbox{which gives}
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t & = & \frac{N}{{\displaystyle\int\limits} p(x)\, dx}
\end{eqnarray*}
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& \propto & \exp \left\lbrace - \sum^K_{k=1} \lambda_k t \right\rbrace
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\end{eqnarray}
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\stepcounter{subsection}
{\newpage
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\samepage \begin{displaymath}p(x) = \frac{1}{2} (1 + ax) \hspace{2em} -1 \le x \le 1
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\langle x^m \rangle & = & \frac{a}{2} \int\limits^{+1}_{-1} x^{m+1} dx \\ 
& = & \frac{1}{m+2} \hspace{2em} : m\ \ \mbox{odd}
\end{eqnarray*}
}

{\newpage
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{\newpage
\clearpage
\samepage \begin{displaymath}a^*_m = (m+2) \sum^N_{i=1} \frac{x^m_i}{N}
\end{displaymath}
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{\newpage
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{\newpage
\clearpage
\samepage \begin{eqnarray*}\sigma^2 \big\vert_m & = & \frac{(m+2)^2}{N} \left\lbrace
\int\limits^{+1}_{-1} x^{2m} (1+ax) dx - \frac{a^2}{(m+2)^2} \right\rbrace \\ 
& = & \frac{1}{N} \left\lbrace \frac{(m+2)^2}{2m+1} - a^2
\right\rbrace
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\sigma^2 \big\vert_1 = \frac{1}{N} (3-a^2)
\end{displaymath}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}\frac{1}{\lbrack N\sigma^2 \rbrack} \big\vert_{m.l.m.} = 
\int\limits^{+1}_{-1} \frac{dx}{p} \left( \frac{\partial p}{\partial a}
\right)^2
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\frac{1}{\lbrack N\sigma^2 \rbrack} \big\vert_{m.l.m.}
& = & \frac{1}{2} \int\limits^{+1}_{-1} dx \frac{x^2}{(1+ax)} \\ 
& = & \frac{1}{2a^2} \left\lbrace \ln \left\lbrack \frac{1+a}{1-a}
\right\rbrack - 2a \right\rbrace \\ 
& = & \frac{1}{3} + \frac{a^2}{5} + \frac{a^3}{7} + \ldots \\ 
& = & \frac{1}{3}\ \mbox{when}\ a=0 
\end{eqnarray*}
}

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{\newpage
\clearpage
\samepage \begin{displaymath}\frac{1}{\lbrack N\sigma^2 \rbrack}\big\vert_{m.l.m.} = \infty\ \mbox{when}\
a=\pm 1 .
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\sigma^2 \big\vert_{m.l.m.} & = & \sigma^2 \big\vert_1 \hspace{1em} a
= 0 \\ 
& < & \sigma^2 \vert_1 \hspace{1em} a\neq 0 .
\end{eqnarray*}
}

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{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 2287
W(\theta , \varphi ) & = & \sum_{lm} a_{lm} Y_{lm} \\ 
a_{lm} & \propto & \langle Y^*_{lm} \rangle
\end{eqnarray*}
}

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{\newpage
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\samepage \psfig{figure=fig13.phys.ps,clip=true,width=3.5in}
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\stepcounter{subsection}
{\newpage
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\end{equation}
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{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 2325
P(x,y) & = & p(y) \frac{\lbrack 1 + \alpha (y) x \rbrack}{2} \\ 
\mbox{Take a $y$-bin} && y_1 \le y \le y_2 \\ 
\mbox{set}\ S(y) = 1: && y_1 \le y \le y_2 \\ 
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\samepage \begin{displaymath}% latex2html id marker 5958
\alpha = 3\hspace{10pt} \frac{{\displaystyle\sum^{N_T}_{i=1}} f(x_i , y_i 
)}{{\displaystyle\sum^{N_T}_{i=1}} g(x_i , y_i )} = \frac{3F}{G} 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}f(x , y) & = & x S(y) \\ 
g(x , y) & = & S(y)
\end{eqnarray*}
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{\newpage
\clearpage
\samepage \begin{equation}\chi^2 = \sum^N_{i=1} \frac{(t_i - e_i)^2}{\sigma^2_i}
\end{equation}
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\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 2367
\frac{\partial\chi^2}{\partial\alpha} & = & \sum^N_{i=1} 2 (t_i - e_i)
\frac{\partial t_i}{\partial\alpha}/\sigma^2_i \\ 
\frac{\partial^2 \chi^2}{\partial\alpha^2} & = & S_1 + S_2 \\ 
S_1 & = & \sum^N_{i=1} 2( t_1 - e_i ) \frac{\partial^2
t_i}{\partial\alpha^2}/\sigma^2_i \\ 
S_2 & = & 2 \sum^N_{i=1} \frac{\left( \frac{\partial
t_i}{\partial\alpha}\right)^2}{\sigma^2_i}
\end{eqnarray*}
}

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\samepage \begin{displaymath}\mu = \sum^N_{i=1} \mu_i
\end{displaymath}
}

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\samepage \setbox\sizebox=\hbox{$\sigma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline6048: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \begin{displaymath}\sigma^2 = \sum^N_{i=1} \omega_i^2
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}% latex2html id marker 6032
\frac{2(t_i - e_i)}{\sigma^2_i}\ \frac{\partial^2 t_i}{\partial\alpha^2}
\end{displaymath}
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\samepage \begin{displaymath}% latex2html id marker 6054
\frac{\partial^2 \chi^2}{\partial\alpha^2} = S_2 = 2 \sum^N_{i=1}
\left( \frac{\partial t_i}{\partial\alpha} \right)^2 / \sigma^2_i
\end{displaymath}
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{\newpage
\clearpage
\samepage \begin{displaymath}-\ln L = - \sum^N_{i=1} \ln q_i
\end{displaymath}
}

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\samepage \begin{displaymath}% latex2html id marker 6083
\frac{-\partial^2 \ln L}{\partial\alpha^2} = S_1 + S_2 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 2411
S_1 & = & \sum^N_{i=1} \frac{1}{q_i} \frac{\partial^2
q_i}{\partial\alpha^2} \\ 
&& \\ 
S_2 & = & \sum^N_{i=1} \frac{1}{q_i^2} \left( \frac{\partial
q_i}{\partial\alpha} \right)^2
\end{eqnarray*}
}

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\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 2427
\langle S_1 \rangle & = & N \left\langle \frac{1}{q}\, \frac{\partial^2
q}{\partial\alpha^2}\right\rangle \\ 
& = & N \int\limits q \cdot \frac{1}{q}\, \frac{\partial^2 q}{\partial\alpha^2}
d\alpha \\ 
& = & N \frac{\partial^2}{\partial\alpha^2} \left\lbrace \int\limits q\,
d\alpha \right\rbrace \\ 
& = & N \frac{\partial^2}{\partial\alpha^2} (1) = 0
\end{eqnarray*}
}

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{\newpage
\clearpage
\samepage \begin{eqnarray}% latex2html id marker 2458
F_j = \frac{\partial\chi^2}{\partial\alpha_j} & = & \sum^N_{i=1}
\frac{(t_i - e_i)}{\sigma^2_i}\, \frac{\partial t_i}{\partial\alpha_j}
\nonumber \\ 
M_{jk} = \frac{1}{2}\, \frac{\partial^2 \chi^2}{\partial\alpha_j
\partial\alpha_k} & = & \sum^N_{j,k=1} \frac{1}{\sigma^2_i}\,
\frac{\partial t_i}{\partial\alpha_j} \,\frac{\partial t_i}{\partial\alpha_k}
\end{eqnarray}
}

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{\newpage
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\samepage \begin{displaymath}% latex2html id marker 6140
\chi^2 (\alpha ) = \chi^2 (\alpha_0) + \sum_j (\alpha-\alpha_0)_j\, F_j
+ \sum_{jk} (\alpha -\alpha_0)_j\, (\alpha -\alpha_0)_k\, M_{jk}
\end{displaymath}
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$\Delta\alpha_j = \alpha_j -
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{\newpage
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\samepage \begin{equation}% latex2html id marker 2488
\chi^2 ({\bf $\alpha$}) = \chi^2 (\Underline{$\alpha$}_0) +
\Delta{\bf $\alpha$ F} + \Delta\Underline{$\alpha$}^T M
\Delta{\bf $\alpha$}
\end{equation}
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{\newpage
\clearpage
\samepage \begin{eqnarray}% latex2html id marker 2495
\frac{\partial\chi^2 ({\bf $\alpha$})}{\partial\alpha_j} & = & 0
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M\Delta{\bf $\alpha$} & = & - \Underline{$F$}
\end{eqnarray}
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\samepage \begin{displaymath}% latex2html id marker 6141
{\bf $\alpha$} = \Delta\Underline{$\alpha$} + \Underline{$\alpha$}_0 
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{\newpage
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\samepage \begin{displaymath}% latex2html id marker 6180
\chi^2 ({\bf $\alpha$}) = \chi^2 (\Underline{$\alpha$}_0) +
\Delta{\bf $\alpha$}^T M\Delta\Underline{$\alpha$}
\end{displaymath}
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{\newpage
\clearpage
\samepage \begin{eqnarray}% latex2html id marker 2521
L & = & \exp \left( - \frac{1}{2} \chi^2 \right) \nonumber \\ 
L & = & L (\alpha_0) \otimes \exp \left( - \frac{1}{2}
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\end{eqnarray}
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\samepage \begin{displaymath}% latex2html id marker 6181
\left\langle (\alpha^* - \tilde{\alpha})_j (\alpha^* - \tilde\alpha
)_k \right\rangle = M^{-1}_{jk}
\end{displaymath}
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\samepage \begin{equation}\sigma_j = \sqrt{(M^{-1})_{jj}}
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\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 2569
\chi^2 & = & \chi^2_0 + \mbox{linear terms} \\ 
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$\Delta\alpha_2 \sim
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\samepage \begin{displaymath}% latex2html id marker 6272
\chi^2 \left( {\bf $\alpha$}_0 + \Delta\Underline{$\alpha$} \right)
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\samepage \begin{displaymath}% latex2html id marker 6298
\left({\bf $\alpha$} - \Underline{$\alpha$}_0 \right)^T \mbox{\bf
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\samepage \begin{displaymath}\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n \ge 0
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{\newpage
\clearpage
\samepage \begin{eqnarray*}\Delta\beta_j & = & \mbox{Predicted value}\ j \le L \\ 
& = & 0 \hspace{1em} j>L
\end{eqnarray*}
}

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{\newpage
\clearpage
\samepage \begin{displaymath}\lambda_L \ge r\lambda_1 \ \ \mbox{and} \ \ \lambda_{L+1} < r\lambda_1
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{\newpage
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\samepage \begin{displaymath}{\bf M} \rightarrow {\bf M} + QI
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{\newpage
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\samepage \psfig{figure=fig20.phys.ps,clip=true,width=4.0in}
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\left( {\bf $\beta$}_0 + \Delta\beta \right) > \chi^2 \left(
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\end{displaymath}
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{\newpage
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\samepage \begin{displaymath}\frac{1}{n}\, T\, r\, {\bf M} = \frac{1}{n}\, \sum^n_{j=1} \lambda_j
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{\newpage
\clearpage
\samepage \begin{equation}p(\chi^2) = \frac{\left( \frac{\chi^2}{2} \right)^{\frac{k}{2} - 1} \exp
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{\newpage
\clearpage
\samepage \begin{eqnarray*}\chi^2 & = & \frac{(x-t)^2}{\sigma^2} \ \mbox{where} \\ 
p(x) & = & \frac{1}{\sigma\sqrt{2\pi}} \exp \left( -
\frac{(x-t)^2}{2\sigma^2} \right) \\ 
p(\chi^2 ) & = & p(x) \frac{dx}{d\chi^2} = \frac{p(x)\sigma^2}{2(x-t)}
\cdot 2 \\ 
p(\chi^2 ) & = & \frac{\exp \left( - \frac{\chi^2}{2} \right)}{\sqrt{2\pi}\ 2\
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{\newpage
\clearpage
\samepage \begin{displaymath}\int\limits^{+\infty}_{-\infty} \exp (i\,t\,\chi^2)\, p(\chi^2)\, d(\chi^2) =
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\clearpage
\samepage \begin{eqnarray*}F(\chi^2) & = & \int\limits^{\chi^2}_0 p(\chi^2) d(\chi^2) \\ 
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e^{\frac{-u}{2}} du
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\samepage \begin{eqnarray*}H^{(0)}\ \mbox{is hypothesis that mean} & = & \mu^{(0)} \\ 
H^{(1)}\ \mbox{is hypothesis that mean} & = & \mu^{(1)} 
\end{eqnarray*}
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\samepage \begin{eqnarray*}t & = & \frac{\left( \overline{x} - \mu^{(0)}\right) 
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\overline{x} & = & \frac{1}{n}\ {\displaystyle \sum^n_{i=1}} x_i \\ 
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\samepage \begin{displaymath}p(t) = \frac{\Gamma \left( \frac{f+1}{2} \right)}{\sqrt{n\pi}\ \Gamma
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\samepage \begin{displaymath}p(x) \propto \exp \left( - \frac{(x-\mu)^2}{2\sigma^2} \right)
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{\newpage
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\pi = \int\limits^{+1}_{-1} \int\limits^{+1}_{-1} dx\, dy\, \theta\, 
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{\newpage
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\samepage \begin{eqnarray*}m & = & 2^{31} \\ 
a & = & 1103515245 \\ 
c & = & 12345
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{\newpage
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{\newpage
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{\newpage
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{\newpage
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&&\exp \left( - \frac{1}{2} \left[ g_1^2 + g_2^2 \right] \right) dg_1\, dg_2 \\ 
& = & \exp \left( - \frac{1}{2} r^2 \right) r\, dr\, d\theta \\ 
& = & \exp \left( - \frac{1}{2} r^2 \right) d\left(\frac{1}{2} r^2
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$x_i$ & 0 & $= F(x_1)$ \\ 
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$x_1 + 2\delta x$ & $p(x_1) + p(x_2)$ & $= F(x_1 + 2\delta x)$ \\ 
$\vdots$ \\ 
$x_{m+1}$ & ${\displaystyle\sum^m_{i=1}} p(x_i)$ & = 1 if normalized 
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\end{document}
