\documentclass[11pt,twoside]{article} \usepackage{latex-macros/course}

%\showsolutions                 % !! use to force solutions & skip query
%\hidesolutions                 % !! use to force no solutions & skip query

\begin{document} 
\inclassproblems{11}

\input{variance1}
\input{variance2}
\input{variance3}
\input{variance4}
\input{variance5}
\input{variance6}
\input{variance7}
\input{variance8}
\input{variance9}
\input{variance10}
\input{variance11}
\input{variance12}
\input{variance13}
\input{variance14}
\input{variance15}
\input{variance16}
\input{variance17}
\input{variance18}

\input{variance20}
\input{variance21}
\input{variance22}
\input{variance23}
\input{variance24}

\instatements{
\appendix

\section{Events}

\begin{eqnarray*}
\pr{\lgunion_{n \in \naturals} A_n} & = & \sum_{n \in \naturals} \pr{A_n}
\mbox{ for pairwise disjoint } A_n\\
%\item $\pr{A-B}  & = &  \pr{A} - \pr{A \intersect B}$
%
% \item $\pr{A \intersect B}  \le  \pr{A}, \label{ab<a}$
%
\pr{\bar{B}}  & = &  1 - \pr{B} \label{1-b}\\
\pr{A \union B}  & = &  \pr{A} + \pr{B} - \pr{A \intersect B}\\
\pr{A \union B} & \leq & \pr{A}+ \pr{B} \quad \mbox{[Boole's inequality]} \\
\pr{A} & \le & \pr{A \union B} \quad \text{[monotonicity]}\\
%
%\item $\pr{\lgintersect_n A_n} \ge 1 - \sum_n \pr{\overline{A_n}}$
%
\prcond{A}{B}  & = &  \frac{\pr{A \intersect B}}{\pr{B}}\\
%\prcond{A}{B} & = & \frac{\prcond{B}{A}\pr{A}}{\pr{B}} quad \mbox{[Bayes Rule]}\\
\end{eqnarray*}

\section{Law of Total Probability}
Let $B_0, B_1, \ldots$ be disjoint events whose union is the
entire sample space.  Then for all events $A$,
\[
\pr{A} = \sum_{i \in \naturals} \pr{A \cap B_i}.
\]

\section{Independence}
\begin{definition*}
Events $A$ and $B$ are independent iff
\[
\pr{A \intersect B} = \pr{A}\pr{B}.
\]

Events $A_0,A_1,A_2,\dots$ are \emph{mutually independent} iff for all
subsets $J \subset \naturals$,
\[
\pr{\lgintersect_{i\in J} A_i} = \prod_{i \in J} \pr{A_i}.
\]

\end{definition*}
}

\end{document}
