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\hypersetup{pdftitle={lecture},
  pdfsubject={A mathematics lecture},
  pdfauthor={Dev Sinha, University of Oregon},
  pdfkeywords={pdftex, acrobat},
  pdfpagemode={FullScreen},
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}

\begin{document}

\begin{quote} 
{Probability is nothing but common sense reduced to
calculation.} \pz

{Pierre Simon Laplace} \pz
\end{quote}

\begin{center}
\includegraphics[width=3in]{Figures/Laplace.jpg}
\end{center}

\newpage

Last time we started to talk about probability as ratio of areas.  \pz
For example, the probability that you would hit a section of a dartboard at
random is the ratio of the area of that section to to the total area of the
dartboard.  \pz  But areas can ``abstractly'' represent probabilities as well.  We start
with a fun example before moving on to a more standard ones. \pb

\begin{example}
You and your housemates have a two-bathroom house, but only one hot shower can
be taken at a time.  You have one housemate whose first class is at the same
time as yours, and unfortunately you both like to take ten-minutes showers
and start your day in the same half-hour 
window. \pz  What are the chances that you get to have an uninterrupted hot shower? 
\pb
\end{example}


{\Large Continuous random variables} \pz

Let's first step back and review and expand our language as developed
for probability within the context of there being a finite number
of outcomes; we will see that this language applies
when the outcomes can vary continuously. \pz

\begin{definition}
\begin{itemize}
\item A  {random} phenomenon is one which is unpredictable but
follows a regular distribution after large numbers of repetitions. \pz
\item A  {random variable} $X$ represents the outcome of a random
phenomenon. \pz
\item The  {sample space} of a random variable is the set of
possible values for the random variable. \pz
\item The  {probability distribution} of a random variable is the
assignment of probabilities to the values in the sample space. \pz
\end{itemize}
\end{definition}
\newpage

\begin{example}
\begin{itemize}
\item Our random phenomenon is throwing two dice. \pz
\item Our random variable $X$ will be the total shown on the two dice. \pz
\item Our sample space is $\{2,3,4,5,\dots  ,12 \}$. \pz
\item Our probability distribution is

\noindent \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$X$ &  2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12   \\ \hline
Probability & .028  & .056 &  .083 & .111 & .139 & .167 & .139 & .111 & .083 & .056 & .028\\ \hline \pz
\end{tabular}
\end{itemize}
\end{example}

So, for example, a ``7'' is six times as likely as a ``2'' or a ``12''. \pz

\begin{definition}
A random variable is  {discrete} if the sample space is a finite
number of values.  It is  {continuous} if the sample space is a
\emph{range} of values. \pz
\end{definition}

The example above is a discrete random variable - why? \pb

\begin{example}
Let $X$ be a randomly chosen number
between $0$ and $3$.  \pz
\begin{itemize}
\item What is our random event? %choice
 \pz our random variable? \pz %value
\item What is $P (0\leq X \leq 3)$? \pz  $P (1.5  \leq X \leq 2)$? \pz
\item What is $P (1\leq X \leq 1.4 \mbox{ or } 2\leq X \leq 2.5)$? \pz
\end{itemize}
\end{example}

In this example $X$ is called  {uniformly distributed} between $0$ and $3$.

\pagebreak

We can redo this example using 
a  {uniform} probability density function. \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 previous example.
\begin{itemize}
\item Our sample space is the closed interval $[0,3]$. \pz
\item Our probability distribution 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}
\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. \pz

{\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

\end{document}

\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?
\end{enumerate}
\end{example}


{\Large  Statistical Estimation} \pz

With our understanding of probability \pz - plus one huge
fact we will soon cover - \pz we can now go back to the important
central question of with what certainty we may draw conclusions
from random surveys and samples of data. \pz

Recall the  {population} refers to the total population we wish to
study and the  {sample} refers to a particular (usually much
smaller) group of the population we have information on. \pz

Suppose we wish to measure the mean
amount people spend purchasing a car in the U.S.  We call this a
 \emph{parameter}.  We won't have enough information to calculate this
number.  \pz
Instead we may interview 1000 people, and get a  \emph{statistic} based
on that 1000 people.  Eventually, we would like some idea how well our
statistic approximates our parameter. \pz

One of the most common parameters we want to measure is the mean
of an entire population,
which we now denote by $\mu$.  \pz  The associated statistic, 
which is the mean of some sample, will be denoted our
usual $\overline{x}$. \pz

\begin{definition}
\begin{itemize}
\item A  {parameter} is  some feature of your  {population} that
you are interested in. \pz
\item A  {statistic} is some feature of your  {sample} with
which you would like to approximate the parameter. \pz
\end{itemize}  
\end{definition}

One of the basic facts of probability and statistics
is that if one takes larger and larger samples
from a population, a statistic will approach its
corresponding parameter (so for example $\overline{x}$
will approach $\mu$).  \pz
When formalized this  is called the ``Law of Large Numbers.''  \pz
But this does not help us in our ultimate goal of extrapolating
data \pz  It says for example that we approach the
right answer if we can survey larger and larger random
samples of car buyers. \pz   But if we can only sample 1000
people, then what we really want to know is how
close we are to the actual parameter ``in all likelihood.'' \pb

\end{document}
 