\setcounter{ExampleCounter}{1}
\begin{itemize}
\item Random variables are denoted with capital letters: 
\vspace{0.5in}

\item Values that the random variable can take are denoted with lowercase letters: 
\vspace{0.5in}

\item The probability that $x$ occurs is written 
\vspace{0.5in}
\end{itemize}

\begin{proc}{Probability Distribution Function}
\vspace*{1.5in}
\end{proc}

\begin{example}{Rolling Two Dice}
Roll two dice and let $X$ be the sum of the values.  Build the probability distribution function for this experiment.\\

Sample space:
\begin{align*}
S = \{&(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{align*}

The sum can be anything from 2 to 12:
\vspace{0.8in}
\end{example}
\vfill
\pagebreak

\begin{example}{Flip a Coin Twice}
Flip a coin twice; let $X$ be the number of heads.  Build the probability distribution function for this experiment.\\

Sample space:
\vspace{0.5in}

Probability distribution function:
\vspace{1.5in}
\end{example}

\begin{example}{College Classes}
There are 5000 undergrads at a college.  The following frequency table describes how many of them are taking a given number of courses.

\begin{center}
\begin{tabular}{c | c}
Number of Courses & Frequency\\
\hline
1 & 478\\
2 & 645\\
3 & 568\\
4 & 1864\\
5 & 1357\\
6 & 88
\end{tabular}
\end{center}

If $X$ is the number of courses that a randomly chosen student is taking, find the probability distribution for $X$.
\vspace{2.5in}

Notice that the probability column is exactly what you would get if you put a relative frequency column on the frequency table.
\end{example}

\begin{formula}{Two Rules}
\begin{enumerate}
\item \text{}
\vspace{0.5in}

\item \text{}
\vspace{0.75in}

\end{enumerate}
\end{formula}

\begin{example}{Supermarket Customers}
The number of customers in line at a supermarket express checkout counter is a random variable with the following probability distribution.
\begin{center}
\begin{tabular}{c | c c c c c c}
$x$ & 0 & 1 & 2 & 3 & 4 & 5\\
\hline
$P(x)$ & 0.10 & 0.25 & 0.30 & 0.20 & 0.10 & 0.05
\end{tabular}
\end{center}
\begin{enumerate}
\item Find $P(2)$.
\vspace{0.75in}

\item Find $P(\textrm{no more than } 1)$.
\vspace{0.75in}

\item Find the probability that no one is in line.
\vspace{0.75in}

\item Find the probability that at least three people are in line.
\vspace{0.75in}

\end{enumerate}
\end{example}
\vfill
\pagebreak



