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\title[Stat 609 Lecture 1]{Stat 609: Mathematical Statistics \\ Lecture 1}
\author[UW-Madison]{Jun Shao}
\institute[Statistics]{%
%  Department of Statistics\\
%  University of Wisconsin\\
%  Madison, WI 53706, USA\\
%~~~\\
  }
\date[2015]{}

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\AtBeginSection[]{\frame{\frametitle{Outline}\tableofcontents[current]}}

\begin{document}

%\frame{\titlepage}

\frame {\vspace{-2mm}
	\frametitle{Stat 609: Mathematical Statistics I\\ (Fall Semester, 2016)\\
		Introduction}
	\vspace{-1mm}
	\begin{alertblock}{Course information}
		\begin{tabular}{llll}
			Instructor & Professor Jun Shao & TA & Mr. Han Chen \\
			Office & 1235A MSC & & 1335 MSC \\
			Phone & 608-262-7938 & & 608-263-5948 \\
			Email & shao@stat.wisc.edu & & hanchen@stat.wisc.edu \\
			Office hours & ??? & & ???  \\
			& or by appointment \\
		\end{tabular}
		
		\vspace{5mm} 
		\begin{tabular}{ll}
			Lecture &  1:00-2:15 MW (SMI 331) \\
			Discussion & 1:00-2:15 F (CHAMBERLIN 2112) \\
			& 2:30-3:45pm F (VILAS 4014) \\
			& Discussions taught by TA \\
		\end{tabular}
	\end{alertblock}
}
\frame{
	\begin{alertblock}{Textbook}
		\begin{tabular}{ll}
			Book & Statistical Inference, 2nd edition \\
			Authors & George Casella and Roger L.\ Berger\\
			Publisher & Brooks/Cole Cengage Learning \\
			Topics covered  & Chapters 1-6 \\
			Syllabus & See the list of topics and schedule\\
			& \\
			Recommended  & Mathematical Statistics (1977, Bickel and Doksum)  \\
			reading books & An Introduction to Probability Theory and \\
			& Mathematical Statistics (1976, Rohatgi) \\
		\end{tabular}
	\end{alertblock}
	\begin{block}{Homework Assignments}
		\begin{itemize}
			\item
			In each lecture, there are 2-4 homework problems selected from the textbook.
			\item
			Problems are assigned in groups with a specified due date.
			\item
			Some or all problems are graded by the TA and a solution will be provided.
		\end{itemize}
	\end{block}}
	\frame{\vspace{-2mm} 
		\begin{alertblock}{Exam Schedule (may be changed)}
			\begin{tabular}{ll}
				1st exam & 1:00-2:15pm on Oct 12, 2016  \\
				2nd exam& 1:00-2:15pm on Nov 16, 2016 \\
				Final exam& 2:45-4:45pm on Dec 22, 2016?  \\
				Note & All exams are closed-book \\
				& You may bring 1 sheet (2 pages) of notes \\
			\end{tabular}
		\end{alertblock}
		\begin{block}{Course Grades}
			\begin{tabular}{lllll}
				Assignments & 20 points & & Grades & A (90-100) \\
				1st exam & 20 points & &  & AB (80-89) \\
				2nd exam & 20 points & & & B (70-79) \\
				Final exam & 40 points & &  &BC (60-69) \\
				Total & 100 points & & & C(50-59) $\cdots\cdots$ \\
				\multicolumn{5}{c} {Adjustments may be made}
			\end{tabular}
		\end{block}
		\begin{alertblock}{Course Material}
			Covered topics, schedules, homework assignments, and lecture slides can be found in my website http://www.stat.wisc.edu/\~{}shao/ 
		\end{alertblock}
		}
	\frame{
		\begin{alertblock}{Required Mathematics}
			\vspace{-5mm}
		\end{alertblock}
		\vspace{-5mm}
		\begin{block}{Linear algebra}
			\begin{itemize}
				\item
				Vectors $c = (c_1,...,c_k)$, row or column, transpose $c'$
				\item
				Matrices $A$, its dimension, transpose $A'$
				\item
				Vector/matrix addition and multiplications: $c'A$, $A+B$, $AB$, ...
				\item
				Positive and non-negative definite matrices and their simple properties.
			\end{itemize}
		\end{block}\vspace{-2mm}
		\begin{alertblock}{Calculus, advanced calculus or mathematical analysis}
			\begin{itemize}
				\item
				Functions (univariate or multivariate), domain and range
				\item
				Limits (as $n \rightarrow \infty$ or as $x \rightarrow x_0$), $\epsilon$-$N$ or $\epsilon$-$\delta$ arguments
				\item
				Continuity and differentiability of functions
				\item
				Right-continuous, left-continuous
				\item
				Right-derivative, left-derivative
				\item
				Integration with finite or infinite integral limits
				\item
				Existence of an integral: e.g., $\int_{-\infty}^\infty |g(x)| dx = \infty$ or a finite number?
			\end{itemize}
		\end{alertblock}
				}
		
		\frame{\vspace{-2mm}
			\begin{alertblock}{Calculus, advanced calculus or mathematical analysis}
				\begin{itemize}
					\item
					Series: when does $\sum_{n=1}^\infty a_n $ converge?
					\item
					Geometric series: \vspace{-2mm}
					\[ \frac{1}{1-x} = \sum_{t=0}^\infty x^t \quad \quad |x| <1 \vspace{-2mm} \]
					\item
					Power series: \vspace{-2mm}
					\[ e^x = \sum_{t=0}^\infty \frac{x^t}{t!} \quad \quad |x| < \infty  \vspace{-2mm} \]
					\item
					Infinity
					\begin{itemize}
						\item For any real number $x$, $\infty + x = \infty$, $x \, \infty =
						\infty$ if $x>0$, $x \, \infty = - \infty$ if $x<0$, and $0 \,
						\infty = 0$;
						\item $\infty + \infty = \infty$;
						\item $\infty^a = \infty$ for any $a>0$;
						\item $x/0 = \infty $ if $x>0$ and $x/0 = -\infty$ if $x<0$;
						\item $\infty - \infty$ or $\infty / \infty$ or $0/0$ is not defined.
					\end{itemize}
				\end{itemize}
			\end{alertblock}\vspace{-2mm}
			\begin{block}{Mathematical induction}
				Prove a statement involving $n$ is true for any fixed $n=1,2,3,...$\\
				(but not $n \rightarrow \infty$)
			\end{block}
		}
		\frame{
			\begin{alertblock}{Knowledge in probability and mathematical statistics}
				\begin{itemize}
					\item
					It is not required, but is very helpful if you took a course in probability and statistics previously
					\item
					Sets, subsets and set operations
				\end{itemize}
			\end{alertblock}
			\begin{block}{Purposes of Stat 609-610}
				\begin{itemize}
					\item
					Learning statistical concepts and knowledge (together with Stat 601-602 or 671-672 sequence)
					\item
					Training at the level of the Mater of Science in Statistics
					\begin{itemize}
						\item
						Stat 609: Basic probability tools useful in statistics (Chapters 1-5) and data reduction principles (Chapter 6)
						\item
						Stat 610: Basic topics in statistical inference (Chapters 7-11)
					\end{itemize}
					\item
					Training in derivations and simple proofs
					\item
					Preparation for a PhD level study in statistics
				\end{itemize}
			\end{block}}
			\frame{\vspace{-2mm}
				\begin{alertblock}{Suggestions for studying}
					\begin{itemize}
						\item
						Take class notes wisely.
						\item
						Do not be afraid of asking questions to the instructor and TA, either during or after the class.
						\item
						Read the relevant part of the textbook carefully after each lecture.
						\item
						Do the related homework problems after each lecture. Do not delay the work until the due date or the day before the due date.
						\item
						Do homework problems independently. Group discussions are encouraged, but you should do your own work.
						\item
						If you have time, try to do more problems from the textbook, which contains many good problems but I don't want to assign all.
						\item
						Study the graded problems from the TA, and the solution. It is a good idea to study a different solution to a (complicated) problem.
						\item
						Review the material before each exam (there will be a review class prior to each exam) and carefully prepare the one sheet notes for each exam.
						\item
						Menage the time wisely in the exam.
					\end{itemize}
				\end{alertblock}}

\frame {
\frametitle{Chapters 1-3: Probability and Random Variable\\
Lecture 1: Sample space and probability}
\vspace{-2mm}
\begin{alertblock}{Random experiment}
An experiment (in a general sense) that results in one of more than one outcomes
with uncertainty in which outcome will be the realization
\end{alertblock}\vspace{-2mm}
\begin{block}{Examples of random experiments}
\begin{itemize}
\item
Tomorrow's weather: Sunny, Cloudy, Rainy
\item
Tomorrow's temperature: $(-M , M)$
\item
Number of car accidents in a city: 0, 1, 2, 3, ...
\item
The life time of a television set: $(0, \infty )$
\item
Toss a coin: Heads or Tails?
\end{itemize}
\end{block}\vspace{-2mm}
\begin{alertblock}{Statistical inference}
\begin{itemize}
\item
Treat data from the experiment as a sample from some population
\item
Make inference about the population
\item
Find things having regular patterns, not appear by chance
\end{itemize}
    \end{alertblock}
}
\frame{
\begin{alertblock}{Probability theory}
A basic tool for statistics, which explains a random experiment in terms of frequencies and their characteristics
\vspace{2mm}

For example, some past observations (data) may tell us that tomorrow's weather is sunny with 80\%, cloudy with 15\%, and rainy with 5\%.
\vspace{2mm}

To study statistical inference, we must have a very good knowledge in probability theory.
\end{alertblock}
\vspace{-2mm}
\begin{block}{Definition 1.1.1. Sample space (or outcome space)}
The set $S$ containing all possible outcomes is called the sample space
    \end{block}
\vspace{-2mm}
\begin{alertblock}{Example}
$S = \{Heads, Tails\}$, $= \{0, 1\}$, $= \{0, 1, 2, 3, ... \}$,
$ = (-M,M)$, $= {\cal R}$ (all real numbers), $= (0 , \infty )$
\vspace{2mm}

Discrete sample space: $S$ contains integers (finitely or infinitely many)
\vspace{2mm}

Continuous sample space: e.g., $S = [0,1]$, $ {\cal R}$, or ${\cal R}^2$.
\end{alertblock}
}
\frame{
\begin{block}{Subsets or sets}
In probability and statistics, we are interested in subsets of a given sample space $S$
(sometime called sets for simplicity).
\vspace{2mm}

$A \subset S$: $A$ is a subset of $S$.
\vspace{2mm}

Example: the life time of a television set is between 0 and 100 is a subset $A = (0, 100]$ of the sample space $S = (0, \infty )$.
\vspace{2mm}

Two very special subsets:
\begin{itemize}
\item
Empty set: $\emptyset $ is the subset of $S$ containing no element.
\item
The sample space $S$ itself is a subset of $S$ (the notation $A \subset S$ allows the special case of $A = S$).
\end{itemize}
\end{block}
\begin{alertblock}{Expressions of subsets}
A subset is  expressed in words or math formulas inside of $\{ \ \ \}$.
\vspace{2mm}

Example:\\
$A = \{ x: \mbox{$x$ satisfies some condition}\}$\\
$A= \{ x: x \in {\cal R} \mbox{ and } x \geq 0 \} = [0, \infty )$
\end{alertblock}
}
\frame{ \vspace{-2mm}
\begin{alertblock}{Relationship}
\begin{itemize}
\item
$A\subset B$: $  x \in A \Rightarrow x \in B$
\item
$A=B $: $A \subset B$ and $B \subset A$
\item
Disjoint: $A$ and $B$ have no common elements
\end{itemize}
\end{alertblock}\vspace{-2mm}
\begin{block}{Example}
$A = \{ \mbox{all even integers} \}$\\
$B = [2, \infty )$
\vspace{2mm}

Obviously, for any $x \in A$, $x \geq 2$ and thus $x \in B$. Hence $A \subset B$.
\end{block}\vspace{-2mm}
\begin{alertblock}{A more complicated example}
Let \vspace{-2mm}
\[ A = \{ x \in {\cal R}^k: x' x \leq c \} \vspace{-2mm}\]
where $c>0$ is a given constant, $x'$ is the transpose of $x$, and $a'b$ is the usual vector product,
and\vspace{-2mm}
\[ B= \left\{ x \in {\cal R}^k:  (y' x)^2 \leq c y' y \ \ \mbox{for all $y \in {\cal R}^k$}\right\}\vspace{-2mm}\]
We want to show $A=B$.
\end{alertblock}
}
\frame{
\begin{alertblock}{A more complicated example}
First, we show $A \subset B$.\\ For any $x \in A$, $x' x \leq c$ and by the Cauchy-Schwartz inequality,
\[  (y' x)^2 \leq x' x y' y \leq c (y' y) \]
for all $y \in {\cal R}^k$. \\
This means $x \in B$ and thus $A \subset B$.
\vspace{2mm}

Next, we show $B \subset A$. \\
If $x \in B$,
\[ (y' x)^2 \leq c y' y \ \ \mbox{for all $y \in {\cal R}^k$} \]
In particular, $y=x$ should satisfy this inequality, which means
\[ (x' x)^2 \leq c x' x \quad \Rightarrow \quad x' x \leq c  \]
This means $x \in A$ and thus $B \subset A$.
\end{alertblock}
}
\frame{\vspace{-2mm}
\begin{block}{Set operation}
\begin{itemize}
\item
The union of $A$ and $B$: $A \cup B = \{ x\!\! : x \in A \mbox{ or } x \in B\}$
\item
The intersection of $A$ and $B$: $ A  \cap B = \{ x\!\! :  x  \in A
\mbox{ and } x \in B\}$

Disjoint $A$ and $B$: $A\cap B = \emptyset $ (the empty set)
\item
The complement of $A$: $A^c = \{ x: x \not \in A\}$, $S^c = \emptyset$
\item
Difference of $A$ and $B$: $A-B = A\cap B^c$
\end{itemize}
\end{block}\vspace{-2mm}
\begin{alertblock}{Venn Diagram}
	\vspace{-12.3cm}
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=10in, width=5.2in]{figure1-1.pdf}
%\centerline{\psfig{figure=vinn1.eps,height=2.1in,width=2.1in}}
\end{center}
\end{figure}
\end{alertblock}
}
\frame{
\begin{block}{Properties}
\begin{itemize}
\item
$A \cap \emptyset = \emptyset $
\item
$A \cup \emptyset = A$
\item
$A \cap S = A$
\item
 $ A \cup S = S$
\item
$\emptyset \subset A \cap B \subset A \ \mbox{or} \ B \subset A \cup B \subset S $
\item
$A \subset B \Rightarrow A \cup B = B$ and $A \cap B = A$
\item
$A -B = A- A\cap B$
\item
$A = (A\cap B )\cup (A\cap B^c)$
\item
$A\cup B = A\cup (B-A) = A \cup (B-A\cap B)$
\item
$ \cup_{i=1}^\infty A_i = A_1\cup A_2 \cup \cdots = \{ x: x \in A_i$ for some $i$ $ \}$
\item
$ \cap_{i=1}^\infty A_i = A_1\cap A_2 \cap \cdots = \{ x: x \in A_i$ for all $i$ $ \}$
\item
$ B \cap \left(\cup_{i=1}^\infty A_i\right) = \cup_{i=1}^\infty (B\cap A_i)$
\item
$B \cup \left(\cap_{i=1}^\infty A_i\right) = \cap_{i=1}^\infty (B\cup A_i)$
\item
DeMorgan's laws:
$\left(\cup_{i=1}^\infty A_i\right)^c = \cap_{i=1}^\infty A_i^c \ $ and $ \ 
\left(\cap_{i=1}^\infty A_i\right)^c = \cup_{i=1}^\infty A_i^c$
\end{itemize}
\end{block}
}
\frame{
\begin{alertblock}{Subset occurrence}
If $x$ in a subset $A$ is the realization of an experiment, then we say that $A$  occurs.
\vspace{2mm}

Let $A$, $B$, $C$ be 3 subsets.\\ Their occurrence can be expressed by the following subsets:
\begin{itemize}
\item
$A$ occurs: $A$
\item
All subsets occur: $A\cap B\cap C$
\item
At least one subset occurs: $A\cup B\cup C$
\item
Only $A$ occurs: $A\cap B^c\cap C^c$
\item
Exactly one subset occurs: $(A\cap B^c\cap C^c) \cup (A^c\cap B\cap C^c )\cup (A^c\cap B^c\cap C)$
\item
None occurs: $A^c\cap B^c \cap C^c$
\item
Not all subsets occur: $A^c \cup B^c \cup C^c$
\item
At least two subsets occur: $(A\cap B )\cup (A\cap C)\cup (B\cap C)$
\item
At most one subset occurs: $(A\cap B^c\cap C^c) \cup (A^c\cap B\cap C^c )\cup (A^c\cap B^c\cap C)\cup (A^c\cap B^c \cap C^c)$
\end{itemize}
\end{alertblock}
}
\frame {
    \begin{alertblock}{Definition 1.2.1}
     A collection  ${\cal F}$ of subsets of a
sample space $S$ is called a $\sigma$-field (or
$\sigma$-algebra) iff it has the following properties:\\
(i) The empty set $\emptyset \in {\cal F}$;\\
(ii) If $A \in {\cal F}$, then the complement $A^c \in {\cal F}$;\\
(iii) If $A_i \in {\cal F}$, $i=1,2,...$, then their union $\cup A_i
\in {\cal F}$.
    \end{alertblock}\vspace{-2mm}
\begin{block}{}
\begin{itemize}
\item
If $A \in {\cal F}$, then $A$ is called an event.
\item
Why do we need to consider $\sigma$-fields? \\
When $S$ is continuous, there are some ``bad" subsets for which a logical definition of measure or probability is not possible.
\item
${\cal F}$ contains good subsets we deal with, but they have to satisfy (i)-(iii).
\item
If $S$ is discrete, we do not need this concept, i.e., 
${\cal F} =$ all subsets of $S$. 
\item
In ${\cal R}^k$, the $k$-dimensional Euclidean space
(${\cal R}^1 = {\cal R}$ is the real line), 
we consider ${\cal F} $ to be the smallest $\sigma$-field
containing all open sets, and sets in ${\cal F}$ are called Borel sets. 
\end{itemize}
\end{block}
}
\frame { \vspace{-2mm}
\begin{alertblock}{How to define the probability of an event $A$?}
\begin{itemize}
\item
Intuitive definition: the rate or percentage of the occurrence of $A$
\item
Statistical definition: if we repeat the same experiment $M$ times, the probability
of $A = $(the number of times $A$ occurs)$/M$ when $M$ is sufficiently large.
\item
Mathematical definition: probability is a special measure.
\item
A measure is an abstract extension of length, area, volume, ...
\end{itemize}
\end{alertblock}\vspace{-2mm}
     \begin{block}{Definition 1.2.4}
A set function $\nu$
defined on a $\sigma$-field ${\cal F}$ is called a {\em
measure} iff it has the following properties.\\
(i) $0 \leq \nu(A) \leq \infty$ for any $A\in {\cal F}$.\\
(ii) $\nu ( \emptyset ) =0$.\\
(iii) If $A_i \in {\cal F}$, $i=1,2,...$, and $A_i$'s are disjoint,
i.e., $A_i \cap A_j = \emptyset$ for any $i \neq j$, then
\[ \nu\left( \bigcup_{i=1}^\infty A_i \right) = \sum_{i=1}^\infty \nu(A_i)
.\vspace{-2mm}  \]
If $\nu ( S )=1$, then $\nu$ is a probability and we use notation $P$ instead of $\nu$.
\end{block}
}

\end{document}
