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\def\header{{\hfill Exam 5
\hfill YELLOW
\hfill A. Miller
\hfill Fall 2005
\hfill Math 210\hfill}}

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       \string& #1 \string&
       \string\qquad \string\\ \string\hline}}

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    \string& \qquad \string\\ \string\hline}
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\begin{document}

\bigskip

{ \bf Show all work.

You do not have to simplify your answer unless asked to do so.

Circle your answer. }

\bigskip

No books and no notes.  You may use a calculator but no other  electronic
devices.


\par\vskip .25in
\begin{center}
Name\makebox[3in]{\hrulefill} \\
\end{center}

\bigskip
Circle your Discussion Section:
\begin{verbatim}

   343  T   12:05--12:55        1412 STERLING
   344  R   12:05--12:55        1327 STERLING
   345  T   13:20--14:10        1327 STERLING
   346  R   13:20--14:10        1327 STERLING 

\end{verbatim}
\bigskip

\begin{center} \Large
 \begin{tabular}{||c|c|c||} \hline\hline
  Problem & Points & Score \\  \hline\hline
   \input{exam.tbl}
  \hline \hline
 \end{tabular}
\end{center}

\immediate\openout\examTBLFile=exam.tbl

\bigskip

Solutions will be posted shortly after the exam:
www.math.wisc.edu/$\sim$miller/m210

\setcounter{page}{0}

\newpage

\prob{12} % 3.1
The three Axioms for a Probability Measure:

\bigskip\bigskip
A probability measure assigns to each event $E$ of a sample space $S$
a number denoted by $Pr[E]$ and called the probability of $E$. This
assignment must satisfy three axioms.


\bigskip

\begin{enumerate}
\item $Pr(E^\prime)=1-Pr(E)\;\;\;$ for any event $E$.
\item $Pr(E_1\cap E_2)=Pr(E_1)Pr(E_2)\;\;\;$ if $E_1$ and $E_2$ are independent
       events
\item $Pr(E_1|E_2)={Pr(E_1\cap E_2)\over Pr(E_2)}\;\;\;$ 
for any events $E_1$ and $E_2$
\item $Pr(E_1\cup E_2)=Pr(E_1)+Pr(E_2)\;\;\;\;$ if $E_1$ and $E_2$ are disjoint
events
\item $E(X+Y)=E(X)+E(Y)$ 
\item $Pr(\emptyset)=0$
\item $Pr(S)=1$
\item $Pr(E_1\cup E_2)=Pr(E_1)+Pr(E_2)-Pr(E_1\cap E_2)\;\;\;$
for any events $E_1$ and $E_2$
\item $0\leq Pr(E)\leq 1\;\;\;$ for any event $E$
\item $Var(X)=E(X-\mu)^2$

\end{enumerate}

\bigskip
Choose exactly 3: 

\bigskip
$\;\;\;\;\;\;\;\;\;\;\;\;\;$ \fbox{\phantom{\rule{2in}{1in}}}

\answer 4,7,9

\prob{12} % 4.1-23
There are 3 blue and 5 green balls in a box.  Balls are selected at random
one after another without replacement, until a green ball is selected.
Let $X$ be a random variable whose value is the number of balls drawn.

Find the density function of $X$ and the expected value of $X$.

Simplify your answers. 

\answer \par\bigskip \begin{tabular}{c|c}
k & Pr(X=k) \\
\hline    
1 & 13/56 \\
2 & 15/56 \\
3 & 5/56 \\\
4 & 1/56 \\
\end{tabular}
\par\bigskip $E(X)=1.5$

\prob{12}  % 9.1- 3
Find the total amount of interest (in dollars) earned on an investment
of \$7,000 with a nominal annual interest rate of 6\% compounded
monthly over a period of two years.

You do not have to simplify your answer.  Circle your answer.

\answer 890.12

\prob{12}  % 9.2-21
The Wisconsin Badgers Turnip Company decides to raise funds for expansion
by issuing zero coupon notes which are payable in 3 years.  The
notes sell for \$2000 and are redeemable 3 years latter for
\$2300.  If such a note is purchased and held to maturity, what
is the effective annual percentage yield to the investor?

You do not have to simplify your answer.  Circle your answer.

\answer 4.8\%

\prob{12}   % 9.3-11
Jebediah is buying a new car.  He has \$3000 in
cash and can borrow the rest of the money needed to buy the
car from his Credit Union at a nominal annual interest rate of
12\% compounded monthly.  
He purchases an \$11,000 car by putting down the cash and financing
the rest for 8 equal monthly payments, each made at the beginning of the
month, and starting on the day he buys the car.   
What is the amount of each payment?

You do not have to simplify your answer.  Circle your answer.

\answer 1035.17

\def\true{\hfill $ True\;\;\;\;\; False\;\;\;\;\;\;\;\;\;\;\;\;$} 
\prob{10} % 1.?
Let $U=\{1,2,3,4,5,6,7\}$ be the universal set and $A,B,C$ subsets
of $U$.  Circle True if the formula is true for all $A,B,C$ subsets
of $U$ and Circle False otherwise.

\begin{enumerate}

\item $n(A\cap B)=n(A)+n(B)-n(A\cup B)$ \true
\item $n(A\times B)=n(A) + n(B)$ \true
\item $A\cup (B\cup C)=(A\cup B)\cup C$ \true
\item $A^\prime\cup B=(A\cap B^\prime)^\prime$\true
\item If $n(A)=5$ and $n(B)=4$, then $A\cap B\neq \emptyset$. \true
\item $(A\cup B)^\prime =A^\prime\cap B^\prime$ \true
\item $A\cap(B\cup C)=(A\cup B)\cap (A\cup C)$ \true
\item $n(A)=7-n(A^\prime)$ \true
\item $A\subseteq B$ iff $A^\prime\subseteq B^\prime$ \true
\item $C^{\prime\prime}=C$ \true

\end{enumerate}

\answer TFTTT TFTFT

\finishtbl

\printanswers

\end{document}