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\begin{document}

\centerline{A. Miller \hfill M431 \hfill Final \hfill Spring 98}
\begin{center}
  Do not hand in this sheet. Show all work.
\end{center}

% 1-4
\prob  A student is to answer 7 out of 10 questions in an examination.
\par (a) How many choices has she?
\par (b) How many if she must answer at least 3 of the first 5 questions?
\answer Self-test 1-4, (a) 120 (b) 110

% 2 - 6
\prob Urn A contains 3 red and 3 black balls, whereas urn B contains 4
red and 6 black balls.  If a ball is randomly selected from each urn,
what is the probability that the balls will be the same color?
\answer Self-test 2-6, $1/2$

% 3 - 8
\prob You ask your neighbor to water a sickly plant while you are on
vacation.  Without water it will die with probability $.8$; with
water it will die with probability $.15$.  You are $90\%$ certain
that your neighbor will remember to water the plant.
\par (a) What is the probability that the plant will be alive when you
return?
\par (b) If it is dead, what is the probability your neighbor forgot
to water it?
\answer Self-test 3-8, (a) $.785$ (b) $.372093$

%4 - 3
\prob A coin that when flipped comes up heads with probability $p$ is
flipped until either heads or tails has occured twice. Find the
expected number of flips.
\answer Self-test 4-3, $2+2p-2p^2$

%5 - 4
\prob The random variable $X$ has probability density
function:
$$ f(x)=\left\{\begin{array}{ll}
               ax+bx^2 & \mbox{ if } 0<x<1\\
                 0     & \mbox{ otherwise }
               \end{array}\right.$$
If $E(X)=.6$, find    $\;\;\;\;\;$
(a) $P(X<{1\over 2})$ $\;\;\;\;\;$
(b) $Var(X)$
\answer Self-test 5-4, (a) $.35$ (b) $.06$

%6 - 6
\prob The joint density function of $X$ and $Y$ is
$$ f(x,y)=\left\{\begin{array}{ll}
                xy & \mbox{ if } 0<x<1, 0<y<2\\
                 0     & \mbox{ otherwise }
               \end{array}\right.$$
\par (a) Find the density function of $X$.
\par (b) Find $E[Y]$
\par (c) Find $P(X+Y<1)$
\par (d) Are $X$ and $Y$ independent?
\answer Self-test 6-6, (a) $f(x)=2x$ if $0<x<1$, $f(x)=0$ otherwise
(b) $4/3$ (c) $1/24$ (d) yes

%8 - 2
\prob Suppose that the number of automobiles sold weekly at a certain
dealership is a random variable with expected value 16 and variance 9.
\par (a) Give a lower bound to the probability that next week's sales
are between 10 and 22 inclusively.
\par (b) Give an upper bound to the probability that next week's sales
exceed 18.
\answer Self-test 8-2, (a) $3/4$ (b) $1/2$

%8 - 7
\prob The servicing of a machine requires two separate steps, with
the time needed for the first step being an exponential random variable
with mean $.2$ hour and the time for the second step being an independent
exponential random variable with mean $.3$ hour.   If a repairperson
has $20$ machines to service, approximate the probability that all
the work can be completed in $8$ hours.
\answer Self-test 8-7, $.1074$

\printanswers

\end{document}