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{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Dan Coman\hspace*{.7cm}

\begin{center}
{\bf Mathematics 323: Introduction to Probabilities }
\\
{\bf Spring Semester 1998}
\\
{\bf Final Exam}
\\
{\bf May 4, 1998}
\end{center}

\begin{tabular}{p{18cm}}\hline
This
\rule[0cm]{0cm}{.5cm}Examination
contains 25 questions, worth 6 points each, for a maximum score of 150. Fill in your answers on this cover sheet 
by placing an {\sf X} through one letter for each problem. You may use your own calculator and your own textbook. 
You may also use a summary (an 8.5"x11" sheet of paper with notes in your writing). You may use 
nothing else. You may not pass a calculator, textbook or summary to another person.

\\[1.3ex]
\hline
\end{tabular}
\vspace*{.5cm}

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\vspace{10mm}
%\\[2ex]
{\bf Sign the pledge:}
\\[1ex]
``On my honor, I have neither given nor received unauthorized aid on this
Exam.''

{\bf Signature:} \underline{\mbox{\hspace*{7cm}}}

%\vspace*{.3cm}
\begin{center}     {\bf GOOD LUCK}\end{center}

\newpage

1. Given the moment generating function $M_X(t)=(1-2t)^{-1/2}$, find $E(X^2)$.

(a) 3 \hfil
(b) $\frac{1}{2}$ \hfil
(c) 1 \hfil
(d) 2 \hfil
(e) $\frac{5}{2}$

\vspace{60mm}
2. A poorly done manuscript 100 pages long is found to have 300 misprints. If a page is selected at random,
what is the probability of finding fewer than the expected number of misprints?

(a) $\frac{1}{300}$ \hfil
(b) $e^{1/3}$ \hfil
(c) $8.5e^{-3}$ \hfil
(d) $100\left(\frac{1}{3}\right)^2\left(\frac{2}{3}\right)^{98}$ \hfil
(e) 0

\vspace{60mm}
3. A grocer buys facial tissue once a month. The number of packages that he sells in any month is normal 
with mean 1600 and standard deviation 40. Find the smallest number of packages he can buy at the beginning 
of the month to be $95\%$ sure he does not run out.

(a) 1700 \hfil
(b) 1666 \hfil
(c) 1633 \hfil
(d) 1680 \hfil
(e) 1640

\newpage
4. If $X$ is normal with parameters $\mu=1$ and $\sigma=3$, what is $E(X^2)$ ?

(a) 4 \hfil
(b) 8 \hfil
(c) 6 \hfil
(d) 10 \hfil
(e) 2

\vspace{60mm}
5. Use the normal distribution to approximate the probability of getting at most 25 "fives" in 180 tosses 
of a fair die.

(a) 0.8413 \hfil
(b) 0.2119 \hfil
(c) 0.1024 \hfil
(d) 0.5793 \hfil
(e) 0.1841

\vspace{60mm}
6. An archer hits the bulls-eye in a target $30\%$ of the time. The probability that it will take 13 shots 
to make 9 bulls-eyes is 

(a) $(0.3)^9(0.7)^4$ \hfil
(b) $495(0.3)^9(0.7)^4$ \hfil
(c) $715(0.3)^9(0.7)^4$ \hfil
(d) $715(0.3)^8(0.7)^4$ \hfil
(e) $495(0.3)^8(0.7)^4$

\newpage
7. Events $A$ and $B$ have probabilities 0.5 and 0.4 respectively, and $P(AB)=0.2$. What is $P(A\cup B)$?

(a) 0.6 \hfil
(b) 1 \hfil
(c) 0.7 \hfil
(d) 0.8 \hfil
(e) 0.9

\vspace{60mm}
8. If $f(x)=cx$ for $0<x<1$ and $f(x)=0$ elsewhere, what must $c$ be in order for $f(x)$ to be a 
probability density function?

(a) 2 \hfil
(b) $\frac{1}{2}$ \hfil
(c) 1 \hfil
(d) 3 \hfil
(e) $\frac{1}{3}$

\vspace{60mm}
9. If $P(A)=0.46$, $P(A\,|\,B)=0.4$, $P(AB)=0.36$, then $P(A\cup B)$ is

(a) 1 \hfil
(b) 0.72 \hfil
(c) 0.96 \hfil
(d) 0.64 \hfil
(e) 0.84

\newpage
10. A bag contains 3 coins: 2 ordinary nickels, and one 2-headed nickel (heads on both sides). One of 
the 3 coins is selected at random and it is flipped. The flip shows a head. What is the probability that 
the 2-headed coin was selected?

(a) $\frac{1}{3}$ \hfil
(b) $\frac{2}{3}$ \hfil
(c) $\frac{1}{2}$ \hfil
(d) $\frac{5}{6}$ \hfil
(e) $\frac{7}{12}$

\vspace{60mm}
11. Say $A$, $B$, $C$ are events for which $P(A)=0.1$, $P(B)=0.2$, $P(C)=0.3$, $P(AB)=0.03$, $P(AC)=0.03$, 
$P(BC)=0.05$. Which of the following is true?

(a) $A$ and $B$ are independent \hfil
(b) $B$ and $C$ are independent \hfil
(c) No two of $A$, $B$, $C$ are independent \\
(d) $A$ and $C$ are independent \quad\quad\quad\quad\quad
(e) You really shouldn't choose this one

\vspace{60mm}
12. Smith and Jones will take turns flipping a fair coin until a head appears. Smith has the first toss. What is 
the probability that Smith tosses the first head?

(a) $\frac{7}{12}$ \hfil
(b) $\frac{5}{6}$ \hfil
(c) $\frac{11}{12}$ \hfil
(d) $\frac{3}{4}$ \hfil
(e) $\frac{2}{3}$

\newpage
13. The number of 5-card poker hands which are four-of-a-kind (out of a deck of 52 cards) is

(a) 13 \hfil
(b) $13\times52$ \hfil
(c) $13^4\times48$ \hfil
(d) 48 \hfil
(e) $13\times48$

\vspace{50mm}
14. A wallet contains 5 pennies and 5 nickels. Three coins are taken at random. Find the probability that
exactly 2 nickels were taken.

(a) $\frac{1}{2}$ \hfil
(b) $\frac{7}{12}$ \hfil
(c) $\frac{2}{3}$ \hfil
(d) $\frac{5}{12}$ \hfil
(e) $\frac{1}{3}$

\vspace{55mm}
15. Let $f_{X,Y}(x,y)=4xy$ for $0\leq x\leq1$, $0\leq y\leq1$, and 0 elsewhere. Find $P(X<12,Y<\frac{1}{2})$.

(a) 1 \hfil
(b) $\frac{1}{4}$ \hfil
(c) $\frac{1}{2}$ \hfil
(d) $\frac{1}{8}$ \hfil
(e) 0

\newpage
16. Let $X$, $Y$ be jointly uniform for $0\leq x\leq2$, $0\leq y\leq3$. Find $P(Y\geq X)$.

(a) $\frac{2}{3}$ \hfil
(b) $\frac{5}{6}$ \hfil
(c) $\frac{5}{12}$ \hfil
(d) $\frac{8}{9}$ \hfil
(e) $\frac{1}{2}$

\vspace{80mm}
17. Let $f_X(x)=\frac{1}{2}x$ for $0\leq x\leq2$, and 0 elsewhere. If $U=X^3$ then, for $0\leq u\leq8$,
$f_U(u)$ is  

(a) $\frac{1}{2}u^{1/3}$ \hfil
(b) $\frac{1}{3}u^{-2/3}$ \hfil
(c) $\frac{1}{6}u^{-1/3}$ \hfil
(d) $\frac{1}{8}u^3$ \hfil
(e) 0

\vspace{60mm}
18. A die is rolled until the first "six" appears. If $X$ denotes the number of the roll on which the 
first "six" appears, then $P(X=3)$ is

(a) $\frac{5}{36}$ \hfil
(b) $\frac{25}{108}$ \hfil
(c) $\frac{25}{216}$ \hfil
(d) $\frac{125}{216}$ \hfil
(e) $\frac{1}{216}$

\newpage
19. If $X$ is exponential with mean $\frac{1}{2}$, then $P(X>7\,|\,X>3)$ is

(a) $e^{-4}$ \hfil
(b) $e^{-7}$ \hfil
(c) $e^{-2}$ \hfil
(d) $e^{-8}$ \hfil
(e) $e^{-3}-e^{-7}$

\vspace{60mm}
20. Suppose $V(X_1-2X_2)=9$, $V(X_1)=3$, $V(X_2)=1$. Find $cov(X_1,X_2)$.

(a) $\frac{1}{2}$ \hfil
(b) $-1$ \hfil
(c) $-\frac{5}{2}$ \hfil
(d) 2 \hfil
(e) $-\frac{1}{2}$

\vspace{60mm}
21. Let $X_1$, $X_2$ be independent, each uniformly distributed on the interval $(-1,1)$. 
If $U=\max\{X_1,X_2\}$ find the density $f_U(u)$ of $U$ for $-1<u<1$.

(a) $\frac{1}{2}$ \hfil
(b) $\frac{3}{8}(u+1)^2$ \hfil
(c) $\frac{1}{2}(u+1)$ \hfil
(d) $u+\frac{1}{2}$ \hfil
(e) $2u+1$

\newpage
22. Let $f(x_1,x_2)=15x_1x_2^2$, for $0\leq x_2\leq x_1\leq1$, and 0 elsewhere, be the joint density of 
$X_1$, $X_2$. Find $P(X_2\geq\frac{1}{4}\,|\,X_1=\frac{1}{2})$.

(a) 0.9 \hfil
(b) 0.8 \hfil
(c) 0.75 \hfil
(d) 0.85 \hfil
(e) 0.875

\vspace{100mm}
23. Given the joint density for $X_1$, $X_2$, $f(x_1,x_2)=1$ for $0\leq x_1\leq1$, $0\leq x_2\leq2$, 
$0\leq2x_1+x_2\leq2$, and 0 elsewhere, find $E(X_1\,|\,X_2=1)$.

(a) $\frac{1}{4}$ \hfil
(b) $\frac{3}{4}$ \hfil
(c) 3 \hfil
(d) 1 \hfil
(e) $\frac{1}{2}$

\newpage
24. Let $X_1$, $X_2$ be jointly uniform on $(0,1)\times(0,1)$. If $U=\sqrt{X_1X_2}$ find the density 
$f_U(u)$ of $U$ for $0<u<1$.

(a) $1-u^2$ \hfil
(b) $2-2u$ \hfil
(c) $-2\ln(1-u)$ \hfil
(d) $-4u\ln u$ \hfil
(e) $3u^2$

\vspace{12cm}
25. Let $X_1,\;X_2,\;\dots,\;X_{75}$ be independent random variables, each of them uniform on $(-1,1)$. 
Using the Central Limit Theorem, approximate $P(X_1+X_2+\dots+X_{75}>0.3)$.

(a) 0.5479 \hfil
(b) 0.4201 \hfil
(c) 0.4582 \hfil
(d) 0.4761 \hfil
(e) 0.5152


\end{document}