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\centerline{Mathematics 375 -- Probability and Statistics 1}
\centerline{Final Examination -- December 17, 2005}
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\noindent
{\it Directions:}  
Do problems I - X in the blue exam booklet.  There are 200 possible 
regular points and 20 Extra Credit points.  Notes:
\item{1)}  If an answer can be expressed in terms of factorials, 
binomial coefficients, the gamma function, etc., it is 
{\it preferable} to leave it in that form.
\item{2)}  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.  
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\noindent
I. 20 students in a statistics class were asked to report the 
number of pets in their families, giving the following data:
$$\matrix{
3, & 3, & 2, & 4, & 1, & 0, & 1, & 2, & 3, & 5,\cr
1, & 2, & 2, & 1, & 1, & 0, & 2, & 4, & 3, & 1\cr}$$
\item{A)} (10) Construct a relative frequency histogram
for this data using one ``bin'' for each integer value.
\item{B)} (10) How many data points are within one sample standard
deviation of the sample mean?  
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\noindent
II.  Suppose that you have a key ring with $N$ keys, exactly
one of which is your house key.  You get home after dark 
and can't see the keys or the ring.  So you try the keys 
in your front door lock one by one until you find the right one. 
\item{A)} (5) Assuming that you 
are not careful and you mix up the keys before each try,
what is the probability that you first
find the correct key on the $y$th try ($y = 1,2,\ldots$)?
\item{B)} (5)  Let $Y_A$ be the discrete random variable
giving the number of the trial on which
you find the correct key by the method from part A.  
What is the expected value of $Y_A$? 
\item{C)} (5) Same question as part A, but now you put aside
each key as you try it so it does not get mixed back in with
the ones you have not tried.  
\item{D)} (5) Let $Y_C$ be the discrete random variable
giving the number of the trial on which
you first find the correct key by the method from part C.  
What is the expected value of $Y_C$?
Which method is superior?  
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\noindent
III. A tea distributor randomly places a miniature porcelain 
figurine of one of 10 different endangered animals in each of its  
boxes of tea.  
\item{A)} (15) What is the probability of getting a
complete set of all 10 animals if you purchase exactly 
10 boxes of the tea?  Explain how you are deriving your answer.
\item{B)} {\it Extra Credit} (10)  What is the expected number
of boxes of tea you need to purchase to obtain a complete
set of all 10 animals?  (Hint:  You get one in the first
box you buy.  What is the expected number of boxes until you 
get a second, different animal?  After you have two, what is the
expected number of boxes to get the third?  etc.)
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\noindent
IV.  
\item{A)} (5) State Bayes' Rule.
\item{B)} (15) According to the American Lung Association, $7\%$ of the 
population has lung disease.  Of those people having
lung disease, $90\%$ are smokers.  Of those not having 
lung disease, $74.7\%$ are nonsmokers.  What is the 
probability that a smoker has lung disease?

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\noindent
V.  The number of customers arriving at a furniture 
store in a one-hour period has a Poisson distribution 
with mean $5$.  
\item{A)}  (10) What is the probability that more than 7 customers
arrive in one hour?
\item{B)}  (10) What is the probability that more than
7 customers arrive in exactly 4 out of the 8 hours 
that the store is open in one day?  (Assume the numbers in
each hour are independent.)  
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\noindent
VI.  The pdf for a continuous random variable $Y$ has 
the form 
$$f(y) = \cases{k(9 - y^2) & if $-3 \le y \le 3$\cr
                   0 & otherwise\cr}$$
\item{A)} (10) Determine the value of $k$.
\item{B)} (15) What are the expected value and variance of $Y$?
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\noindent
VII.  Professional golfers' tee shot distances are normally 
distributed with $\mu = 282.2$ yards and $\sigma = 8.12$ yards.  
\item{A)} (10) What
is the probability that a tee shot goes more than 300 yards?  
\item{B)} (10) What
is the probability that a tee shot goes between 270 and 290 yards?  
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\noindent
VIII. (15) Two phone calls are made independently in a fixed 1-hour
period.  Each call occurs at a random time uniformly distributed
on the interval $[0,1]$.  
 Find the joint density function for the times $Y_1, Y_2$
of the two calls and use it to find the probability that the 
two calls are made within 5 minutes of each other 
(i.e. $|Y_1 - Y_2| \le 5/60$).
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\noindent
IX.  Two random variables $Y_1,Y_2$ have joint density 
$$f(y_1,y_2) = \cases{2 e^{-y_1-y_2} & if $0 \le y_1 \le y_2 < \infty$\cr
                           0 & otherwise\cr}$$
\item{A)} (10) Find $P(Y_2 \le 2 + Y_1)$.  (Draw a picture!)
\item{B)} (10) Find the marginal density $f_1(y_1)$. 
What is the distribution of $Y_1$?  
\item{C)} (10) Let $U = 3Y_1 + 2Y_2$.  Find the mean $\mu_U$.
\item{D)} {\it Extra Credit} (10) Find the variance $\sigma_U^2$ for $U$ from part C.\bigskip
\noindent
X.  
\item{A)} (10) Let $Y$ have a gamma distribution
with $\alpha=3.2$, and $\beta=1$.  Derive a general formula
for $E(Y^k)$, where $k \ge 1$ is an integer.   (See note 1 in the directions!)
\item{B)} (10) Let $Y_1,\ldots,Y_n$ be independent,
all with gamma distributions with the same $\alpha$, and $\beta$.
Using moment-generating functions, 
determine the distribution of the mean 
$\overline{Y} = {1\over n}\sum_{i=1}^n Y_i$.  
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\centerline{\it Have a peaceful and joyous holiday season!}
\bye

