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Mathematics 375 -- Probability Theory\\
Final Examination -- December 16, 2011
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\noindent
{\it Directions}  
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You have {\it as much time as you need\/} to work on this exam.
Do problems I - VIII in the blue exam booklet.  There are 200 possible 
regular points and 20 Extra Credit points.  Notes:
\begin{enumerate}
\item[1)]  If an answer can be expressed in terms of binomial
coefficients, the gamma function, factorials, etc., it is 
{\it preferable} to leave it in that form.
\item[2)] It is OK (even suggested!) to use general formulas we have
derived, etc. rather than working out probability integrals, 
etc. from scratch.
\item[3)]  Even if you 
don't see how to derive a complete solution to a problem, don't leave
it blank.  Partial credit will be given for relevant definitions, 
ideas that could lead to solutions, and so forth.  
\end{enumerate}
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\noindent
I.  A large number of observations of a certain continuous
random variable $Y$ were taken,
and a typical subset of $n=50$ of them were ordered to produce
the data set below:
$$\begin{matrix}
0.10 & 0.23&0.25&0.26&0.26&0.28&0.31&0.39&0.42&0.44\\
0.44 & 0.48&0.53&0.54&0.55&0.66&0.69&0.70&0.72&0.73\\
0.73 & 0.74&0.74&0.76&0.77&0.78&0.81&0.95&0.97&1.02\\
1.03&1.06&1.07&1.13&1.13&1.14&1.19&1.22&1.27&1.28\\
1.30&1.34&1.34&1.37&1.39&1.40&1.43&1.43&1.46&1.48
\end{matrix}$$
\begin{enumerate}
\item[A)] (10) Construct a relative frequency histogram for 
this data using 10 equal bins on the interval $\left[0,\frac{3}{2}\right]$.
\item[B)] (20) 
Two models are proposed for the distribution of $Y$:
\begin{enumerate}
\item[1.] a {\it uniform} distribution on 
$\left[0,\frac{3}{2}\right]$, or 
\item[2.] a distribution described by the 
probability density function (pdf)
$$f(y) = \begin{cases} 
\frac{8}{9}\left(\frac{3}{2} - y\right) & \text{if } 0\le y \le \frac{3}{2}\\
      0 & \text{otherwise}
\end{cases} $$
\end{enumerate}
Find the expected values and variances of the two
proposed model distributions.  
\item[C)] (5) The actual sample mean is
$\overline{y} = .8542$ and the sample standard deviation
is $s \doteq .4033$.  Is either of the models in part B 
clearly more likely as the underlying distribution
from which the values in part A were generated?  Explain.
\end{enumerate}

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\noindent
II.  (15) A bin contains five components from supplier $A$, three
from supplier $B$, and four from supplier $C$.  If three
of the components are selected randomly (without replacement)
for testing, what is the probability that the components chosen
will come from at most two of the suppliers?

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\noindent
III.  Rutabaga seeds from grower $A$ have an $90\%$ germination
rate, while seeds from grower $B$ have a $80\%$ germination rate and 
those from grower $C$ have a $85\%$ germination rate.
A seed packaging company buys $25\%$ of its seeds
from grower $A$, $35\%$ from grower $B$, and $40\%$ from grower
$C$, then mixes them thoroughly in making up packets for sale.
\begin{enumerate}
\item[A)] (10) What is the probability that a randomly 
selected seed will germinate?
\item[B)] (10) Given that a seed did not germinate,
what is the probability that it came from grower $B$?
\end{enumerate}

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\noindent
IV.  A permutation of $A = \{1,2,3,4,5\}$ is a one-to-one, onto mapping
from this set to itself.  For instance, the $f$ defined by 
$$f(1) = 2, f(2) = 4, f(3) = 3, f(4) = 1, f(5) = 5$$
is one such permutation.  Consider a permutation selected at
random, and let $Y$ be the discrete random variable representing
{\it number of elements of the set
$A$ that are mapped to themselves}.  For instance, if the 
$f$ above was selected, then the value of $Y$ is  $2$ 
(since $f(3) = 3$ and $f(5) = 5$).  
For this problem,  you are given the information that the 
moment-generating function of $Y$ is 
$$m(t) = \frac{44}{120} + \frac{45}{120}e^t + \frac{20}{120} e^{2t} + 
\frac{10}{120} e^{3t} + \frac{1}{120} e^{5t}$$
\begin{enumerate}
\item[A)]  (10) Find the mean and variance of $Y$.
\item[B)]  (10) What is the probability that a randomly selected
permutation maps {\it no element of $A$ to itself}?
\end{enumerate}

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\noindent
V.  A candy maker produces candy bars that have a stated label
weight of 57.4 grams.  The manufacturing process is subject to 
some randomness, though, and the actual weights of the bars
are normally distributed with mean $\mu = 58$ grams and $\sigma = 2$ 
grams.
\begin{enumerate}
\item[A)]  (5) What is the probability that a single candy bar 
has weight $ < 57$ grams?  
\item[B)]  (5) If candy bars are selected independently and at random
from the production line, what is the probability that at least
4 trials will be necessary to find one that has weight $< 57$ grams?
\item[C)]  (5) If 20 candy bars are selected independently and at
random from the production line, what is the probability 
that 3 or fewer will have weight $< 57$ grams?
\item[D)]  (10)  The production process can be changed to alter
the average weight $\mu$ of the bars, but $\sigma = 2$ cannot
be changed.  What value 
for $\mu$ will make the probability that a candy bar weighs
$< 57.4$ grams equal to $.01$?    
\end{enumerate}

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\noindent
VI.  
\begin{enumerate}
\item[A)] (10) State the Central Limit Theorem.
\item[B)] (10) From experience, a charity knows that the 
contributions it receives have mean $\mu = 3125$ and $\sigma = 250$.  
If $n = 2000$ of the contributions are selected at random, 
estimate the probability that the average of those $2000$ 
contributions is $> 3200$.  
\end{enumerate}

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\noindent
VII.   Let $Y$ be a random variable with density function
$$f(y) = \begin{cases} k y^4 e^{-2y} & \text{if } y > 0\\
                      0 & \text{otherwise}
         \end{cases}
$$
\begin{enumerate}
\item[A)] (10) Determine the value of $k$, and find $E(Y)$ and $V(Y)$.
\item[B)] (10) Let $Y_1, \ldots, Y_n$ be independent random variables
all with density as above.  What is the distribution of
$\overline{Y} = \frac{1}{n}(Y_1 + \cdots + Y_n)$?
\end{enumerate}

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\noindent
VIII.
Let $Y_1$ and $Y_2$ be random variables with joint pdf 
$$f(y_1,y_2) = \begin{cases} 
                    3y_1^2 y_2 & \text{if } -1 \le y_1 \le 1, 0 \le y_2 \le 1 \\
                         0 & \text{otherwise}
               \end{cases}
$$
\begin{enumerate}
\item[A)] (10) What is $P(Y_2 > Y_1)$? ({\it Hint:} Draw a picture!)
\item[B)] (10) Are $Y_1$ and $Y_2$ independent?  Why or why not?
\item[C)] (10) Determine $V(3Y_1 + 2Y_2)$.
\item[D)] (10) Let $U = Y_1^2$.  Determine the pdf for $U$.  
\end{enumerate}

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\noindent
{\it Extra Credit.}  (20)
A volcano is located at the center $(x,y) = (0,0)$ of a circular island
1 mile in radius.  When the volcano erupts, it emits
lava that spreads over a random sector of the island.  The diagram
below shows one such sector between two radii drawn with dashes.  
The middle radius of the sector (solid line in diagram) makes 
a random angle with the positive $x$-axis,
$\Theta$, which is uniformly distributed on the 
interval $[-\pi,\pi]$.  The total central angle of the sector (the 
angle between the two dashed radii in the diagram)
is a second random variable $\Phi$ that has pdf
$$f(\varphi) = \begin{cases} c(\pi/2 - \varphi) & \text{if } 0 \le \varphi \le \pi/2\\
                             0 & \text{otherwise}
            \end{cases} $$
for some constant c.  The angles $\Theta$ and $\Phi$ are independent.

\begin{center}
\includegraphics[width=2.5truein,height=2.5truein]{Volcano.ps} 
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You own a villa at $(x,y) = (1,0)$ on the coast of the island.
What is the probability that any single eruption buries your villa 
in lava?
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\emph{Have a peaceful and joyous holiday season!}
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