\documentclass[11pt]{report}
\usepackage{smstc,probability,color,mathrsfs}
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\Streamname{Probability}  
\Unitname{} 
\Thisyear{2007/08}  
\AuthorFt{Takis Konstantopoulos, Heriot-Watt University}

\def\N{\mathbb{N}}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\C{\mathbb{C}}
\def\FF{\mathscr{F}}
\def\SS{\mathscr{S}}
\def\BB{\mathscr{B}}
\def\LL{\mathcal{L}}
\renewcommand{\phi}{\varphi}
\renewcommand{\epsilon}{\varepsilon}
\newcommand{\1}{{\text{\Large $\mathfrak 1$}}}
\renewcommand{\comp}{\raisebox{0.1ex}{\scriptsize $\circ$}}
\newcommand{\cadlag}{{c\`adl\`ag} }
\renewcommand{\var}{\operatorname{var}}
\renewcommand{\cov}{\operatorname{cov}}
\renewcommand{\limsup}{\varlimsup}
\renewcommand{\liminf}{\varliminf}
\renewcommand{\d}{\text{\rm\,d}}
\newcommand{\eqdist}{\stackrel{\text{d}}{=}}
\newcommand{\dirac}{\text{\large $\mathfrak d$}}
\newcommand{\Ri}{\Rightarrow}
\def\iid{{i.i.d.}\xspace}
\def\jid{{i.i.d.}}
\def\ie{{i.e.}\xspace}
\def\eg{{e.g.}\xspace}
%A \xrightarrow[n]{} 0

\parskip \bigskipamount


\begin{document} \initdoc 

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\begin{unit}{1}{Random variables}

A random variable is a measurable function $X$ between two measurable spaces,
an ``abstract'' one $(\Omega, \FF)$ and a ``concrete'' one $(S, \SS)$.
In other words, we require that the inverse image by $X$ of each element 
of $\SS$ be an element of $\FF$.
We denote this, implicitly, by the symbol
\[
X : (\Omega, \FF) \to (S, \SS).
\]
The terms in quotes have nothing to do with Mathematics but, rather, with
our human interpretation of it. For example, we want, most
of the time, to leave $(\Omega, \FF)$ unspecified but 
$(S, \SS)$ can be, for example, $(\R, \BB)$, where $\BB$ is the class of
Borel sets on $\R$. In such a case we call $X$ ONE random variable,
omiting the numeral ONE when not necessary. As another example, let
$(S, \SS) = (\R^2, \BB^2)$, where $\BB^2$ are the Borel sets on $\R^2$.
In this case we refer to $X$ as TWO random variables, because we may,
by choosing Cartesian co\"ordinates on the $\R^2$, represent
$X$ by $(X_1, X_2)$, where $X_1$ is one random variable and $X_2$ is
also one random variable.
More generally, we may let $(S, \SS) =(\R^d, \BB^d)$, and then
we refer to $X$ a random ($d$-dimensional) vector.

The real fun of course starts when $(S, \SS)$ is chosen to be a large 
(but concrete) space, for example the space of real-valued
continuous functions on the real line. In this case, $\SS$ is usually chosen
to be the minimal $\sigma$-algebra containing the open sets of continuous
functions, open with respect to the topology of uniform convergence on 
compacta.
In such a case, $X$ is called random function or stochastic process
(stochastic being a synonym of random and process being a synonym
of function).

Of course, the concept of ``random variable'' has nothing to 
do with randomness. The adjective ``random'' is attached because, 
in addition to the above, we also consider some probability (measure) 
$\Pr$ on $(\Omega, \FF)$ .
As a mathematical definition, the concept of random variable is
purely functional.
So let us do that: let $\Pr$ be a probability on $(\Omega, \FF)$.
The distribution of the random variable $X$ is a probability $\Pr_X$ on
$(S, \SS)$ which is induced, in the most natural fashion, by $X$:
\[
\Pr_X(B) = \Pr(X^{-1}(B)), \quad B \in \SS.
\]





\end{unit}



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\begin{unit}{2}{Joint distributions and independence}
\end{unit}

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\begin{unit}{3}{Important special distributions}
\end{unit}


\end{document}
























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%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Citations  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsubsection{Citations}  %% Done in the usual way:

As shown by \cite[Theorem 5.1]{lubich} \dots

We also refer to \cite{jones.th,Pujols}\dots

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Exercises  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Exercises} %%or \section{Exercises} to number it and add to toc

%% If your unit contains exercises, then put them here.  The exercise
%% environment can be used for this (but note that it doesn't display
%% the word "Exercises")
\begin{exercise} 
\item For each \dots

\item If $f: \R^2 \rightarrow \C$ is defined by \dots
\end{exercise}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%  Bibliography  %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%  Bibliography  %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% Instructions for non-BibTeX users (for a single unit document)
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%% Please use the following syntax to create the bibliography
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\begin{thebibliography}{9}
%% or \begin{thebibliography}{99} if there are >9 of them

%% Book
\bibitem{jones.th}
{\sc D.~S. Jones}, {\em Theory of Electromagnetism}, O. U. P., 1964.

%% Journal
\bibitem{lubich}
{\sc C.~Lubich}, {\em On the multistep time discretization of linear
initial--boundary value problems and their boundary integral
equations}, Numerische Mathematik, 67 (1994), pp.~365--389.

%% Proceedings article
\bibitem{Pujols}
{\sc A.~Pujols}, {\em Time--dependent integral method for {M}axwell
equations}, in Mathematical and Numerical Aspects of Wave
Propagation Phenomena, G.~Cohen, L.~Halpern, and P.~Joly, eds.,
SIAM, 1991, pp.~118--126.

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