% This is a simple LaTex sample document that gives a submission format
%   for IEEE PAMI-TC conference submissions.  Use at your own risk.

% Make two column format for LaTex 2e.


\documentclass[11pt]{article} %
\usepackage[dvips]{graphicx,graphics,color}
\usepackage{times,amsmath,amsfonts}

\setlength{\textheight}{10in} \setlength{\textwidth}{6.875in}
\setlength{\columnsep}{0.3125in} \setlength{\topmargin}{-0.5in}
\setlength{\headheight}{0in} \setlength{\headsep}{0in}
\setlength{\parindent}{1pc}
\setlength{\oddsidemargin}{-.2in}  % Centers text.
\setlength{\evensidemargin}{-.2in}

% Add the period after section numbers.  Adjust spacing.
\newcommand{\Section}[1]{\vspace{-8pt}\section{\hskip -1em.~~#1}\vspace{-3pt}}
\newcommand{\SubSection}[1]{\vspace{-3pt}\subsection{\hskip -1em.~~#1}
        \vspace{-3pt}}
\newcommand{\bqn}{\begin{eqnarray}}
\newcommand{\eqn}{\end{eqnarray}}
\newcommand {\diff}[1] {\frac{\partial}{\partial #1}}
\newcommand{\jacob}[3]{\frac{\partial^2 #3}{\partial #1 \partial #2}}
\newcommand{\der}[2]{\frac{\partial #2}{\partial #1}}
\begin{document}

% Don't want date printed
\date{January 20, 2005}

% Make title bold and 14 pt font (Latex default is non-bold, 16pt)
\title{Stat 324: Lecture 02\\
Probability}

% For single author (just remove % characters)
\author{Moo K. Chung\\
mchung@stat.wisc.edu}

% For two authors (default example)
\maketitle \thispagestyle{empty}
\begin{enumerate}

\item An {\em experiment} is any action whose outcome is subject to
uncertainty (randomness). The {\em sample space} of an experiment is
the set of all possible outcomes of the experiment. An {\em event}
is a subset of the sample space. Two events are {\em mutually
exclusive} if they have no outcomes in common.

\item Given a sample space $S$,
probability, denoted by $P$, is a function that satisfies the
following three conditions: (i) $P(A) \geq 0$, (ii) P(S) = 1, (iii)
For mutually exclusive events $A_1, A_2 \in S$ $P(A_1 \cup A_2) =
P(A_1) +P(A_2)$. Then we have the following properties

\begin{enumerate}

\item If $A, B \in S$ and $A \subset B$, $P(A) \leq P(B)$.

\item $P(A) = 1 - P(A^c)$.

\item If $A$ and $B$ are mutually exclusive, then $$P(A \cap B)=0.$$

\item $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
\end{enumerate}


\item Prove $$P(A \cap B) \geq P(A) + P(B) - 1,$$
when $A,B$ are two events in the same sample space.\\

{\it proof.} We have shown in the class that
$$P(A \cup B) = P(A)+P(B)-P(A \cap B).$$ But $P(A \cup B) \leq 1$. So by
rearranging the terms, $P(A \cap B) \geq P(A) + P(B) -1$. This
inequality is called {\it Bonferroni inequality}.

\item Show $P(A_1 \cup A_2 \cup A_3) \leq P(A_1) + P(A_2) + P(A_3)$.\\

{\it proof.} $P(A_1 \cup B) = P(A_1) + P(B) - P(A_1 \cap B) \leq
P(A_1) + P(B)$. Now let $B=A_2 \cup A_3$. Then similarly $P(B) \leq
P(A_2) + P(A_3)$. So $P(A_1 \cup A_2 \cup A_3) \leq P(A_1) + P(B)
\leq P(A_1) + P(A_2) + P(A_3)$.

\item For events $A$ and $B$, find formulas for the probabilities
of the following events in terms of the quantities $P(A),P(B)$
and $P(A \cap B)$ only.\\
(i) either $A$ or $B$ but not both \\
(ii) at most one of $A$ or $B$

{\it solution.} (i) $P((A \cup B)\backslash(A \cap B))=P(A \backslash B) + P(A \backslash B) = P(A)+P(B)-2P(A\cap B).$ \\
(ii) Let $D$ be the event of {\it at most one of $A$ or $B$}. The
complement of $D$ is the event of {\it both $A$ and $B$}, i.e. $D^c=
A \cap B$. So $P(D)= 1-P(D^c)= 1-P(A \cap B).$


\end{enumerate}

{\bf Assigned Problems} Read Chapter 2. Exercise 2.16., 2.20.
Homework I. Submit assigned problems in Lecture 1-3 on January 27
9:30AM. No late assignment will be accepted.

\end{document}