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\begin{center}
{\LARGE \bf Chapter 1} \\
{\LARGE Elements of Stochastic Processes}\\
\end{center}

\section{Review of Basic Terminology and Properties of Random Variables and Distribution Functions}
\par A random variable $X$ is \textit{discrete} if there exists a finite or numerable set of distinct values
$\lambda_i$ for $i\in\mathbb{Z}$ s.t $\pr{X=\lambda_i}=a_i>0$ and $\sum_i a_i = 1$.
If on the other hand $\pr{X=\lambda}=0$ for any $\lambda$, the variable is called \textit{continuous}.

\par \textit{Distribution function} $F$ is what we normally called the cumulative distribution function (CDF),
 and it is defined as

\begin{equation}
    F(\lambda) = \int_{-\infty}^{\lambda}p(t)dt
\end{equation}

where $p(t)$ is the \textit{probability density}. $p$ may or may not exist for continuous random variables. For
discrete variables this boils down to
\begin{equation}
    F(\lambda) = \pr{X\leq\lambda}.
\end{equation}

\par \textit{Moments} for random variables are given by

\begin{equation}
    E[X^m] = \int_{-\infty}^{\infty}x^mp(x)dx
\end{equation}
for a moment of order $m$. This integral turns into a sum for discrete variables. The first moment is the \textit{mean},
$m_X$. The second central moment (the 2nd moment of random variable $X - m_X$ is the \textit{variance}, $\sigma_X^2$.

\par Functions of random variables are random variables. Expectations of functions of random variables can be easily
computed.

\subsection{Joint Distribution Functions}
The joint distribution of a pair of discrete random variables $X, Y$ is
\begin{equation}
    F(\lambda_1, \lambda_2) = \pr{X\leq\lambda_1, Y\leq\lambda_2}.
\end{equation}

The marginal distribution of $X$ is then $F(\lambda, \infty) = \lim_{\lambda_2\to\infty}F(\lambda, \lambda_2)$.
The marginal distribution of $Y$ is defined similarly.
The corresponding probability density, if it exists, for a joint distribution would then be

\begin{equation}
    F(\lambda_1, \lambda_2) = \int_{-\infty}^{\lambda_1}\int_{-\infty}^{\lambda_2}p(s,t)dsdt.
\end{equation}

\par If $X, Y$ have respective means $m_X, m_Y$, their covariance is

\begin{equation}
\sigma_{XY} = E[(X-m_X)(Y-m_Y)].
\end{equation}

\par If $X_1, X_2$ are independent with respective distributions $F_1, F_2$, the distribution $F$ of their sum
is

\begin{equation}
F(x) = \int F_1(x-y)dF_2(y) = \int F_2(x-y)dF_1(y).
\end{equation}
The same convolution goes for the density functions if they exist.

\subsection{Characteristic Functions}