\setcounter{ExampleCounter}{1}
\begin{itemize}
\item Random variables are denoted with capital letters: \[X\]
\item Values that the random variable can take are denoted with lowercase letters: \[x\]
\item The probability that $x$ occurs is written \[P(X=x) = P(x)\]
\end{itemize}

\begin{proc}{Probability Distribution Function}
A table that lists each possibility and its probability.

\paragraph{Ex:} Roll a die; let $X$ be the number that comes up
\begin{center}
\begin{tabular}{r | c c c c c c}
$x$ & 1 & 2 & 3 & 4 & 5 & 6\\
\hline
& & & & & & \\
$P(x)$ & $\dfrac{1}{6}$ & $\dfrac{1}{6}$ & $\dfrac{1}{6}$ & $\dfrac{1}{6}$ & $\dfrac{1}{6}$ & $\dfrac{1}{6}$
\end{tabular}
\end{center}
\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:
\begin{center}
\begin{tabular}{r | c c c c c c c c c c c}
$x$ & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12\\
\hline
& & & & & & & & & & & \\
$P(x)$ & $\dfrac{1}{36}$ & $\dfrac{2}{36}$ & $\dfrac{3}{36}$ & $\dfrac{4}{36}$ & $\dfrac{5}{36}$ & $\dfrac{6}{36}$ & $\dfrac{5}{36}$ & $\dfrac{4}{36}$ & $\dfrac{3}{36}$ & $\dfrac{2}{36}$ & $\dfrac{1}{36}$
\end{tabular}
\end{center}
\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:
\[S = \{HH,\ HT,\ TH,\ TT\}\]

Probability distribution function:
\begin{center}
\begin{tabular}{c | c}
$x$ & $P(x)$\\
\hline
0 & 0.25\\
1 & 0.5\\
2 & 0.25
\end{tabular}
\end{center}
\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 $X$.
\begin{center}
\begin{tabular}{c | c}
$x$ & $P(x)$\\
\hline
1 & 0.096\\
2 & 0.129\\
3 & 0.114\\
4 & 0.373\\
5 & 0.271\\
6 & 0.018
\end{tabular}
\end{center}

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 The probabilities are all between 0 and 1
\item The probabilities add up to 1:
\[\sum P(x) = 1\]
\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 $P(2)$.
\[P(2) = 0.3\]

\item Find $P(\textrm{no more than } 1)$.
\[P(0) + P(1) = 0.10 + 0.25 = 0.35\]

\item Find the probability that no one is in line.
\[P(0) = 0.1\]

\item Find the probability that at least three people are in line.
\[P(3) + P(4) + P(5) = 0.20 + 0.10 + 0.05 = 0.35\]

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



