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\title[Stat 609 Lecture 3]{Stat 609: Mathematical Statistics \\ Lecture 3}
\author[UW-Madison]{Jun Shao}
\institute[Statistics]{%
%  Department of Statistics\\
%  University of Wisconsin\\
%  Madison, WI 53706, USA\\
%~~~\\
  }
\date[2014]{}

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

\begin{document}

%\frame{\titlepage}

\frame {
\frametitle{
Lecture 3: Measurable space, measure and probability}
\vspace{-2mm}
    \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}{}<2->
\begin{itemize}
\item
$(S, {\cal F})$ is a measurable space if ${\cal F}$ is a $\sigma$-field on $\Omega$\\
\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.
\end{itemize}
\end{block}
}

\frame {
\vspace{-2mm}
\begin{block}{Two trivial examples}
\begin{itemize}
\item
${\cal F}$ contains $\emptyset$ and $S$
only
\item
${\cal F}$ contains all subsets of $S$ (power set)
\end{itemize}
\end{block}
\vspace{-2mm}
\begin{alertblock}{Why do we need to consider other $\sigma$-field?}

We may be interested in a particular collection of sets

e.g., ${\cal F} =  \{ \emptyset , A, A^c , S \}$, where $A \subset
S$

${\cal C} =$ a collection of sets of interest

${\cal C}$ may not be a $\sigma$-field

$\sigma({\cal C})$: the smallest $\sigma$-field
containing ${\cal C}$ (the $\sigma$-field generated by ${\cal
C}$)

$\sigma({\cal C})= {\cal C}$ if ${\cal C}$ itself is a $\sigma$-field

 $\sigma( \{ A \})= \sigma( \{ A, A^c\}) = \sigma( \{ A, S \})
= \sigma( \{ A, \emptyset \}) = \{ \emptyset , A, A^c , S \}$
\end{alertblock}
\vspace{-2mm}
     \begin{block}{Borel $\sigma$-field}<2->
 ${\cal R}^k$: the $k$-dimensional Euclidean space
(${\cal R}^1 = {\cal R}$ is the real line)

${\cal O}$: the collection of all open sets

${\cal B}^k = \sigma ( {\cal O})$: the Borel $\sigma$-field on ${\cal R}^k$

${\cal B}^k = \sigma ( {\cal C})$, ${\cal C} $ is the collection of all closed sets

$C \in {\cal B}^k$, ${\cal B}_C= \{ C \cap B: B \in {\cal B}^k\}$
is the Borel $\sigma$-field on $C$
     \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 case of measure.
\item
A measure is an abstract extension of quantities like Length, area, volume, ...
\end{itemize}
\end{alertblock}\vspace{-2mm}
     \begin{block}{Definition 1.2.4}<2->
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
\[ \nu\left( \bigcup_{i=1}^\infty A_i \right) = \sum_{i=1}^\infty \nu(A_i)
.  \]
\end{block}

}

\frame {
\begin{alertblock}{Definition 1.2.4}
\begin{itemize}
\item
$(S , {\cal F}, \nu )$ is a measure space.
\item
If $\nu ( S )=1$, then $\nu$ is a probability measure.
\item
We usually use notation $P$ instead of $\nu$.
\item
$(S , {\cal F}, P)$ is a probability space.
\end{itemize}
\end{alertblock} \vspace{-2mm}
\begin{block}{Two important examples of measures}<2->
\begin{itemize}
\item
Counting measure:

Let $S$ be a discrete sample space, ${\cal F}$ the collection of all
subsets, and $\nu(A)$
 the number of elements in $A \in {\cal F}$ ($\nu(A)=\infty$ if $A$
contains infinitely many elements).

Then $\nu$ is a measure on
${\cal F}$ and is called the {\em counting measure}.
\item
Lebesgue measure:

There is a unique measure $m$ on $({\cal R}, {\cal B})$ that
satisfies $ m([a,b])=b-a$ for every finite interval $[a,b]$,
$-\infty < a \leq b < \infty$.

This is called the {\em Lebesgue
measure}.

If we restrict $m$ to the measurable space $([0,1],{\cal
B}_{[0,1]})$, then $m$ is a probability measure.
\end{itemize}
     \end{block}
}
\frame{\vspace{-3mm}
\begin{block}{Theorem 1.2.6.}
Let $S = \{s_1,s_2,... \}$ and ${\cal F}$ be all subsets of $S$.\\
Let $p_1,p_2,...$ be nonnegative numbers that sum to 1. \\
For any $A \in {\cal F}$, define
\[ P(A) = \sum_{i: s_i \in A} p_i \]
(sum over the empty set is defined to be 0). \\
Then $P$ is a probability on ${\cal F}$.
\end{block}\vspace{-3mm}
\begin{alertblock}{Theorem 1.2.8.}<2->
If $P$ is a probability and $A$ is an event, then\\
a. $P(\emptyset ) =0$;\\
b. $P(A) \leq 1$;\\
c. $P(A^c) = 1- P(A)$.
\end{alertblock}\vspace{-3mm}
\begin{block}{Theorem 1.2.9.}<3->
If  $P$ is a probability and $A$ and $B$ are events, then\\
a. $P(B\cap A^c) = P(B)-P(A\cap B)$;\\
b. $P(A \cup B) = P(A)+P(B) - P(A\cap B)$;\\
c. If $A \subset B$, then $P(A) \leq P(B)$.
\end{block}
}
\frame{
\begin{block}{Example.}
Select randomly 4 cards from a deck 52 cards.\\
Let $A$ be the event that 1 ace and 1 king are selected and $B$ be the event that
2 aces are selected.
\[ P(A) = \frac{{4 \choose 1}{4 \choose 1}{44 \choose 2}}{{52 \choose 4}} = \frac{4 \times 4\times 44 \times 43 /2}{270725} =
0.055909 \]
\[ P(B) = \frac{{4 \choose 2}{48 \choose 2}}{{52 \choose 4}} = \frac{6\times 48\times 47/2}{270725}=0.025000   \]
\[ P(A^c) = 1- P(A) = 1- 0.055909 = 0.944091 \]
\[ P(A \cap B) =  \frac{{4 \choose 1}{4 \choose 2}{44 \choose 1}}{{52 \choose 4}} = \frac{4 \times 6\times 44}{270725} =0.003901 \]
\[ P(A - B) = P(A) - P(A \cap B) = 0.055909 - 0.003901 = 0.052008 \]
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.052008 + 0.025  = 0.077008 \]
\end{block}
}
\frame{\vspace{-2mm}
\begin{block}{Birth date problem}
There are $n$ balls in $N$ boxes, $n \leq N$. \\
What is the probability that there are $n$ boxes having exactly one ball?
\[ P = \frac{N(N-1)\cdots (N-n+1)}{N^n } = \frac{N!}{N^n (N-n)!} \]
When there are $n$ persons in a party, what is the probability $p_n$ that there are at least two persons having the same birth date?\\
$n$ persons $= n$ balls\\
$N = 365$ days
\[ P( \mbox{no one has the same birth date with others} ) = \frac{365!}{365^n (365-n)!}  \]
\[ p_n = 1- \frac{365!}{365^n (365-n)!} \]
$p_{20} = 0.4058$, $p_{30} = 0.6963$, $p_{50}= 0.9651 $, $p_{60} = 0.9922$
\end{block}\vspace{-2mm}
\begin{alertblock}{Reading assignments: Sections 1.2.3 and 1.2.4}<2->
\end{alertblock}
}
\frame{
\begin{block}{Union of three events}
\[ P(A \cup B \cup C) = P(A)+P(B)+P(C)\]
\[  - P(A\cap B) - P(A \cap C)- P(B\cap C) + P(A\cap B\cap C)\]
Example:\\
$P(A) = P(B) = P(C) = 1/4$, $A$ and $B$ are disjoint, $P(B\cap C) = P(A\cap C) = 1/6$\\
$P(A \cup B \cup C) =?$ \vspace{-2mm}
\[ P(A \cup B \cup C)  = 1/4 + 1/4 + 1/4 - 0 - 1/6 - 1/6 + 0 = 7/12 \]
\end{block}\vspace{-2mm}
\begin{alertblock}{General addition formula}<2->
Events $A_1,...,A_n$
\[ P \left( \cup_{i=1}^n A_i \right) = \sum_{i=1}^n P(A_i) - \sum_{ i  < j} P(A_i\cap A_j) \]
\[ + \sum_{i <j <k} P(A_i \cap A_j \cap A_k )  + \cdots +(-1)^{n-1} P(A_1 \cap A_2 \cap \cdots \cap A_n) \]
\end{alertblock}
}
\frame{
\begin{block}{Theorem 1.2.11.}
If $P$ is a probability, $A$, $A_1,A_2,...$, $C_1,C_2,...$ are events, and $C_i \cap C_j = \emptyset$ for any $i \neq j$ and
$\cup_{i=1}^\infty C_i = S$, then \\
a. $P(A) = \sum_{i=1}^\infty P(A\cap C_i)$;\\
b. $P(\cup_{i=1}^\infty A_i ) \leq \sum_{i=1}^\infty P(A_i)$.
\begin{itemize}
\item
Property a is useful when $P(A\cap C_i)$ is easy to calculate.
\item
Property b is called Boole's inequality.
\end{itemize}
\end{block} \vspace{-2mm}
\begin{alertblock}{Bonferroni's inequality}<2->
For $n$ events $A_1,..,A_n$,
\[ 1- P\left( \cap_{i=1}^n A_i\right) = P\left( \cup_{i=1}^n A_i^c \right)  \leq \sum_{i=1}^n P(A_i^c)
= n - \sum_{i=1}^n P(A_i) \]
Then
\[ P\left( \cap_{i=1}^n A_i\right) \geq  \sum_{i=1}^n P(A_i) - (n-1) \]
which is called Bonferroni's inequality.
\end{alertblock}
}


\end{document}
