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\begin{document}

{\Large Probability distributions}\pz

To be systematic about probabilities in the continuous setting, we have to 
formalize the notion of distribution. \pz

\begin{definition}
\begin{enumerate}
\item A distribution for a continuous random variable can be
any non-negative function $D$ where the total area under the function is $1$. \pz
\item If $X$ is distributed according to a distribution $D$ then
$P (a \leq X \leq b)$ is the area between the line $x = a$, the line $x = b$,
the graph of $D$, and the $x$-axis. \pz
\item The sample space for such a variable is all $x$ where $D$ is non-zero. \pz
\item If $D$ is equal to either $0$ or one other fixed number, we say $D$
is a uniform distribution. \pz
\end{enumerate}
\end{definition}

%\includegraphics{Figures/uniform0-3.eps} \pz

\begin{example}
Revisit the example about numbers between 0 and 3. 
\begin{itemize}
\item Our sample space \pz  is the closed interval $[0,3]$. \pz
\item Our probability distribution \pz  is the function 
\[
p (t) = \left\{ \begin{array}{ll}
1/3 & 0 \leq t \leq 3\\
0  & \mbox{else}
\end{array} \right. \pz
\]
\end{itemize}
Again, what is the probability that some randomly
chosen number is between 1.5 and 2.4? \pz Etc. \pz
\end{example}


\begin{example}
Which of the following are distributions?\pz
\begin{itemize}
\item $f(x) = 1$ for $x$ between $-2$ and $-1$, and zero otherwise. \pz
\item $f(x) = 1$ over the whole real number line. \pz
\item $f(x) = \frac{x}{2}$ for $x$ between 0 and 2 and zero otherwise. \pz 
\item $f(x) = \frac{x}{2}$ for $x$ between $-1$ and $1$ and zero otherwise. \pz
\end{itemize}
\end{example}

\begin{example}
Suppose that $X$ is distributed according to 
$$D = \begin{cases} \frac{1}{12} \;\; 1 \leq x \leq 3 \\
\frac{1}{2} \;\; 3 \leq x \leq 4 \\  \frac{1}{9} \;\; 4 \leq x \leq 7 \\
0 \;\; {\text{otherwise}}
\end{cases}$$
\begin{itemize}
\item What is the state space? \pz
\item What is $P(2 \leq X \leq 3)$? \pz
\item Which is more likely, that $X$ is between $3$ and $4$ or that $X$
is between $5$ and $7$?  \pz
\end{itemize}
\end{example}

Compare the answer to the last question to the answer you would get if
$X$ were uniformly distributed. \pb

One way to get a distribution is to take the histogram of a data set and divide
by the total number of individuals so that the area is one.  \pz
The probabilities computed correspond to percentages of the data. \pz

\begin{example}
Translate between probability and percentile questions about 
scores of students taking an exam. \pb
\end{example}



{\Large Normal distributions} \pz

\begin{example}{[Main Example]}
$\mathbf{Z}$ is a number chosen with normal distribution $N (\mu ,\sigma)$. \pz
We'll use $N (0,1)$ as our first example. \pz
%\includegraphics{Figures/normal0-1.eps}
\begin{itemize}
\item Our random event is the choice of a number. \pz
\item Our random variable $\mathbf{Z}$ is the value of the number. \pz
\item Our sample space is all possible real numbers. \pz
\item Our probability distribution is the function 
\[
e^{-t^{2}/2}/\sqrt{2\pi }. \pz
\]
\item What is $P (\mathbf{Z}>3)$? (Hint: use 68-95-99.7 rule.) \pz % = .0015$ by 
\item What is $P (0 \leq Z \leq 2)$?  \pz % can be read off of Table A with a bit of work.
\item $P (\mathbf{Z}>100)$? \pause 
%The probability  $P
%(\mathbf{Z}>22)$ is less that $\frac{1}{10^{99}}$, and will come out
%as $0$ on our calculators.
\end{itemize}
\end{example}

\newpage

Doing calculations with normal distributions works much as before. \pz
Instead of calculating percentiles, we are calculating probabilities. \pz


\begin{example}
Let our random event be the random choice of a U.S. adult male,
assuming a probability distribution which is 
 approximately normal $N (70,4).$. \pz
\begin{enumerate}
% \includegraphics{Figures/normal70-4.pdf}
\item What is our random variable?  \pz Our state space? \pz
(theoretical vs. actual) \pz
\item What is the probability that our randomly chosen man
is between 70 inches and 73 inches tall? \pb
\end{enumerate}
\end{example}



\end{document}
 