

\begin{slide}
\subsection{\bf 1.3\hspace{1em} Continuous Distributions}
\subsection{\bf General (Borel) Formalism}
\begin{itemize}
\item To be formal and reveal relics of a primeval mathematical training: 
take random variables in $\Re_1$ and consider special Borel set $\xi\le x$:
It has probability $F(x)$ with $F(\infty)=1$.

\item Now, in some cases, we can differentiate $F(x)$ and so write:
\[
F(x) = \int\limits^x_{-\infty} p(x) dx
\]
where $p(x)$ is called the probability density.  

\item One can clearly interpret $p(x) dx$ as the probability that
random variable $x$ takes values in range $x$ to $x+dx$.

\item Consider examples:
\begin{description}
\item{$\bullet$ (a)} In the case of tossed coin, let heads be $x=1$ and
tails $x=0$.  Then for (an unbiased) coin
\end{description}
\[
p(x) = \frac{1}{2} \left\lbrack \delta (x) + \delta (x-1) \right\rbrack
\]
\end{itemize}
\end{slide}

