% PDT Regression
% WEEK 1 (JP edit)
%
\renewcommand{\coddate}{{\small \today}}
\renewcommand{\sheweek}{Introductory Material}
%
\begin{center}
\Huge {\bf ST402 \\ Principles and Methods \\
of Statistical Practice}
\\ Weeks 1 to 5: Distribution Theory \\
Michaelmas Term \\ 2006-2007 \\
\vspace{1.5cm} \large Teacher:  Martin Knott B607
\end{center}
\normalsize
%
\vspace{2.5cm}
%
%*******
\section*{Course Aims and Objectives}
%*******
%
This part of the course aims to convey a thorough understanding of
probability and distribution theory. A range of methods, related
to distributions of practical importance, are taught. The course
provides a strong theoretical foundation for MSc Statistics
students.

The following list gives you an idea of the sort of things you
will be able to do by the end of the first five weeks -- it does
not come close to covering everything. By the end of the first
part of the course you should: \vS
\begin{itemize}
\item be able to work out probabilities associated with simple (and
not so simple) experiments,
\item know the
distinction between a random variable and an instance of a random
variable,
\item for any given distribution,
be able to select suitable methods and use them to work out
moments,
\item be familiar with a large number of distributions,
\item understand relationships between variables,
conditioning, independence and correlation,
\item feel at ease with joint distributions and conditional distributions,
\item be able to put together all the theory and techniques you
have learnt to solve practical problems.
\end{itemize}
\newpage

%******************************
\section*{Introductory Material}  \label{intro}
%******************************
%

%*******
\subsection*{Structure of the Course}
%*******
%
Each student should usually attend \textbf{three hours per week} in
the Michaelmas term. This is made up of: \vS
\begin{enumerate}
\item two hour lecture/problem class Tuesday 15:00-17:00 in S300, \vS \item one problem class
Friday 09:00-10:00 in S221, \vS
\end{enumerate}

The course in assessed by a two hour written exam in the summer
term.
%

%---------------------------------------------------------------
%
%*******
\subsection*{Books}
%*******
%
There are a large number of books that cover at least part of the
material in the course. Finding a useful book is partly a question
of personal taste. I suggest you look at what is available in the
library and find a text that covers the material in a way that you
find appealing and intelligible. Reproduced below is the reading
list along with some additional texts that may be worth looking
at.

\textbf{Main texts}
\begin{itemize}
\small
\item Casella, G. and R. L. Berger.
\textit{Statistical inference.}
\href{http://catalogue.lse.ac.uk/cgi-bin/Pwebrecon.cgi?BBID=752288}{[QA276
C33]}

\item Larson, H. J. \textit{Introduction to probability theory and
statistical inference.} [QA273.A5 L33]

\item Meyer, P. L. \textit{Introductory probability and statistical
applications.}
\href{http://catalogue.lse.ac.uk/cgi-bin/Pwebrecon.cgi?BBID=21700}{[QA273.A5
M61]}
\end{itemize}

\normalsize \textbf{Books covering similar material to main texts}
\begin{itemize}
\small
\item Hogg, R. V. and A. T.
Craig. \textit{Introduction to mathematical statistics.} [QA276.A2
H71]

\item Freund, J. E. \textit{Mathematical statistics} [QA276
F88]

\item Hogg, R. V. and E. A. Tanis. \textit{Probability and
statistical inference.} [QA273 H71]

\item Mood, A. M.,
F. A. Graybill and D. C. Boes. \textit{Introduction to the theory of
statistics.} [QA276.A2 M81]

\item Bartoszy\'{n}ski, R. and M. Niewiadomska-Bugaj. \textit{Probability and
Statistical Inference.} [QA273 B29]
\end{itemize}

\normalsize
 \textbf{Worth a look}
 \small
\begin{itemize}
\item Cox, D. R. and D. V. Hinkley. \textit{Theoretical statistics.}
[QA276.A2 C87] \newline (Not great to learn from but a good
reference source.)

\item Stuart, A. and J. K. Ord. \textit{Kendall's Advanced Theory of Statistics}
\textbf{1}, \textit{Distribution theory.} [QA276.A2 K31] \newline
(A bit arcane but covers just about everything.)

\item Grimmett, G. R. and  D. R. Stirzaker. \textit{Probability and random
processes.} [QA273 G86] \newline (Very succinct and very precise.
One for those who like it mathematical.)

\item Johnson, N. L. and S. Kotz (some volumes also with N.
Balakrishnan) \textit{Discrete Univariate Distributions},
\textit{Continuous Univariate Distributions}, etc. [QA273.6 J61]
\newline (The place to go if you have a particular question
about a distribution.)

\item Larsen R. J. and M. L. Marx. \textit{An Introduction to
Mathematical Statistics and its Applications.} [QA276 L33]
\newline (Good introduction to probability. Lots of examples)

\end{itemize}
\normalsize
%----------------------------------------------------------

%-----------------------------------------------------------
%
%*******
\subsection*{Pre-requisites}
%*******
%
The background of students on this course is very varied. The
course does not assume much knowledge of probability or statistics
but we will move very quickly through the material. Below is a,
far from exhaustive, list of mathematical tool that we will be
using. \vS
\begin{itemize}
\item \textbf{Sets}: union, intersection, complement. \vS
\item \textbf{Series}: arithmetic, geometric, Taylor. \vS
\item \textbf{Integration}: standard integrals, integration by
parts. \vS
\item \textbf{Differentiation}: standard differentials, product
rule, function of a function rule.
\end{itemize}
%

%***************
\subsection*{A Guide to Content}
%************
The following is a guide to the content of the course rather than
a definitive syllabus. Throughout the course examples, with
varying degrees of realism, will be used to illustrate the theory.
The material that goes into the exam will be determined by what is
covered in the lectures.
\begin{enumerate}
\item \textbf{Events and their Probabilities}:
Interpretations of probability / Sample space: experiments,
outcomes / Events / Probability Space: probability measure,
permutations and combinations / Conditional Probability:
partitions, law of total probability, Bayes' theorem,
independence.
\item \textbf{Random Variables and their Distributions}: Random
Variables: cumulative distribution function / Discrete Random
Variables: probability mass function / Continuous Random
Variables: probability density function, Lebesgue-Stieltjes
integral / Support and indicators / Mean / Variance / Expectation
Operator: Markov,
 and Chebyshev inequalities / Moments: generating
functions / Application: functions of random variables and simulation.
\item \textbf{Multivariate Distributions}: Joint Distributions:
bivariate distributions, marginal distributions, expectations of
functions of random variables, multivariate generalisation /
Conditional Distribution: conditional mass function, conditional
density function, conditional expectation, conditional variance /
Dependence: independence, covariance, correlation / Application: sums
of independent random variables, random sums / Application:
multivariate normal density, fundamentals of linear prediction / Joint
Moments: joint generating function, conditional generating function /
Application: laws of large numbers and the central limit theorem /
Transformations of random variables.
\end{enumerate}

We will encounter many different distributions during the course.
These include:
\begin{itemize}
\item \textbf{Discrete Distributions}: degenerate, Bernoulli, binomial,
geometric, negative binomial, Poisson, hypergeometric, uniform.
\item \textbf{Continuous Distributions}: uniform, normal,
exponential, chi-square, $t$, $F$, gamma, beta.
\end{itemize}

 \newpage
\subsection*{Course Handouts}
The printed notes given out in lectures are intended as a
\textbf{reference}. They provide key equations and course structure but
are not intended as lecture notes or as a replacement for copying stuff
down from the board. I have made some corrections to these notes but
\textbf{there will still be errors!} If you spot something that looks
strange or wrong, please let me know. The notes, questions and some
solutions are available in the public folders under {\tt
Statistics/ST402}, and in the course handouts boxes in the Statistics
department (6th floor of Columbia House). If you are missing any
handouts, please collect them from one of these three places.
