%ST402 PMSP slides

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ST402\\2003-2004\\ Dr M. Knott, B607\\
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\foilhead{Lecture 1 - Objective and Outcomes }




You will often see $(\Omega, {\cal F}, P)$ used to denote a probability
space. This lecture explains where this comes from and how it relates to
reality.

\begin{itemize}
\item Experiments, Outcomes $\omega$, Sample space $\Omega$.
\item Collections of events ${\cal F}$.
\item Probability measure $P$.
\item Counting outcomes.
\item Bayes' theorem.
\item Independence.
\end{itemize}
\foilhead{}
 After reviewing the notes you should: \small
\begin{itemize}
\item know the distinction between events and outcomes,
\item understand how probability measures correspond to our
intuitive ideas about how likely events are,
\item be able to work out the appropriate method for counting
outcomes for any given situation,
\item understand how conditioning can be reversed using Bayes'
theorem,
\item know a large number of the consequences of events being
independent
\end{itemize}


\foilhead{}
 Sample space for throwing die twice
\begin{tabular}[t]{cccccc}
(1,1) & (1,2) & (1,3) & (1,4) & (1,5) & (1,6) \\
& & & & & \\
 (2,1) & (2,2) & (2,3) & (2,4) & (2,5) & (2,6) \\
& & & & & \\
 (3,1) & (3,2) & (3,3) & (3,4) & (3,5) & (3,6) \\
 & & & & & \\
 (4,1) & (4,2) & (4,3) & (4,4) & (4,5) & (4,6) \\
& & & & & \\
(5,1) & (5,2) & (5,3) & (5,4) & (5,5) & (5,6) \\
& & & & & \\
 (6,1) & (6,2) & (6,3) & (6,4) & (6,5) & (6,6)
\end{tabular}

\foilhead{}
 \textbf{Definition}: collection of events, ${\cal F}$
\newline For any experiment, the events form a collection of subsets of
$\Omega$ which we denote ${\cal F}$. The collection ${\cal F}$ has the
following properties:
\begin{enumerate}
 \item[i.] $\emptyset \in {\cal F}$,
 \item[ii.] if $A \in {\cal F}$ then$A^c \in {\cal F}$,
 \item[iii.] if $A_1,A_2, \ldots, \in {\cal F}$then $\bigcup_{i=1}^{\infty} A_i \in {\cal F}$.
\end{enumerate}

\foilhead{}
 \textbf{Definition}: probability measure, $P$
\newline $P$ on $(\Omega, {\cal F})$ is a function $P:{\cal F}
\longrightarrow [0,1]$ satisfying; \vS
\begin{enumerate}
\item[i.] $P(A) \geq 0$, \vS
\item[ii.] $P(\Omega)=1$, \vS
\item[iii.] if $A_1,A_2,\ldots,$ is a collection of mutually
exclusive members of ${\cal F}$ then
\newline $P(\bigcup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} P(A_i)$.
\end{enumerate}
\foilhead{Properties of probability measures}
\begin{enumerate}
\item[a.] $P(A^c) = 1 - P(A)$. \vS
\item[b.] If $A \subseteq B$ then \newline
$P(B) = P(A) + P(B \backslash A) \geq
P(A)$. \vS
\item[c.] $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. \vS
\item[d.] More generally if $A_1,\ldots,A_n$ are events then \newline
\footnotesize
\begin{tabular}{l}
 $P(\bigcup_{i=1}^n A_i)=\sum_i P(A_i) - \sum_{i<j}
P(A_i \cap A_j)$ \\ $+ \sum_{i<j<k} P(A_i \cap A_j \cap A_k) +
\ldots + (-1)^{n+1} P(A_1 \cap \ldots \cap A_n).$
\end{tabular}
\item[e.] Boole inequality: $P(\bigcup_{i=1}^n A_i) \leq \sum_{i=1}^n
P(A_i)$.
\end{enumerate}


\foilhead{}
 Examples taken from Larsen and Marx, \newline \textit{An
Introduction to Mathematical Statistics}

\textbf{Birthday problems}: What is the smallest number of people
needed to have probability of over 0.5 of: \vS
\begin{enumerate}
\item[a.] at least one person with the same birthday as me? \vS
\item[b.] at least two people with the same birthday as each other?
\end{enumerate}
\foilhead{Poker problems}
 a poker hand is five cards dealt from a
(perfectly shuffled) standard 52 card pack. What is the probability of:

\begin{enumerate}
\item[a.] a full house? \item[b.] a pair?  \item[c.] a straight?
\end{enumerate}


\foilhead{Partitions}
 A set of events $B_1,\ldots,B_n$ defined on
$\Omega$ is a partition if:
\begin{enumerate}
 \item Exhaustive: $\bigcup_{i=1}^n B_i = \Omega$,
 \item Mutually exclusive: $B_i \cap B_j = \emptyset$ for $i \neq j$,
 \item Non-zero probability: $P(B_i)>0$ \newline for $i=1,\ldots,n$.
\end{enumerate}

\foilhead{Some properties of independent events}
\begin{enumerate}
\item \hspace{-0.2cm}\begin{tabular}[t]{l}
If $P(A) > 0$ then  $P(B|A) = P(B) \Longleftrightarrow A \perp B$. \\
If $P(B) > 0$ then  $P(A|B) = P(A) \Longleftrightarrow A \perp B$.
\end{tabular}
\item If $A \perp B$ then $A^c \perp B^c$, $A^c \perp B$ and \newline
$A \perp B^c$.
\end{enumerate}
\end{document}
