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\centerline{Mathematics 375 -- Probability and Statistics I}
\centerline{Final Examination -- December 14, 2009}
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\noindent
{\it Directions}
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Do all work in the blue exam booklet.  There are 200 possible regular points and
10 possible Extra Credit points.  If an answer can be expressed in terms of
binomial or multinomial coefficents, factorials, the gamma function, etc. please leave it in that
form rather than evaluating.  Even if you cannot see how to completely solve a problem,
do not 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.  Twenty students in a probability and statistics class were asked to report
the number of pets in their families, giving the following data:
$$3,3,2,4,1,0,1,2,3,5,1,2,2,1,1,0,2,4,3,1$$
\item{A)} (10)  Construct a relative frequency histogram for this
data using one ``bin'' for each integer value.
\item{B)} (10) How many of the data values are within two sample standard
deviations of the sample mean?  How does this compare with the ``empirical rule?''
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\noindent
II.  Suppose you have a key ring with $N$ keys, exactly one of which is 
your car key.  You are parked in a country lane on a moonless night and can't see the keys or
the ring.  So you try the keys in your car door lock one by one until you
find the right one.
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\item{A)}  (10) Assuming you are not careful and you mix up all the keys after 
each try, what is the probability that you take at least $y$ tries to 
find the right key?
\item{B)} (5)  Let $Y$ be the discrete random variable giving the number of the 
trial on which you find the right key.  What is the expected value of $Y$?
\item{C)} (10)  Now suppose you are more careful and you take each wrong key 
off the key ring and put it aside
after you try it so that it does not get mixed back in with the keys you 
have not tried.  Let $Z$ be the discrete random variable giving the number of the
trial on which you find the right key by this method.  What is the expected value of $Z$?
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\noindent
III. (Hypothetical)  In the city of Worcester, 50 percent of 
registered voters are Democrats, 30 percent are Republicans,
and 20 percent are Independents.  
 In the recent Senate primary, 15 percent of Democrats voted,
 10 percent of Republicans voted, and 35 percent of Independents
 voted.
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\item{A)} (10)  If a single voter is selected at random, what
is the probability that he or she voted in the Senate primary?
\item{B)} (10)  Given that a registered voter did vote, what is
the probability that the voter was an Independent? 
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\noindent
IV. Let $X$ be a random variable for which $E(X) = \mu$ and 
$V(X) = \sigma^2$.  Let $c$ be an arbitrary constant.
\item{A)}  (10)  Show that $E((X - c)^2) = (\mu - c)^2 + \sigma^2$.
\item{B)}  (5)  For which $c$ is $E((X - c)^2)$ a minimum?
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\noindent
V.  The number of customers arriving at a fast food restaurant
in a typical hour has a Poisson distribution with mean $20$. 
\item{A)}  (5) What is the probability that 17 customers
arrive in a given hour?
\item{B)}  (10)  If the store is open 12 hours a day, 
what is the probability that exactly 17 customers
will arrive in exactly 4 out of the 12 hours?
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VI.  A continuous random variable $Y$ has p.d.f 
$$f(y) = \cases{ c(y^2 + y) & if $1 \le y \le 2$\cr
                     0 & otherwise\cr}$$
\item{A)} (5)  What is the value of $c$?
\item{B)} (5)  What is $P(1 \le y \le 3/2)$?
\item{C)} (10) Find $E(Y)$ and $V(Y)$.
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\noindent
VII. (15) The diameters of the bolts in a large box follow
a normal distribution with mean $2$ cm and a standard
deviation of $0.03$ cm.  The diameters of the holes
in the nuts in another large box are also normally
distributed, but with mean $2.02$ cm and standard deviation
$0.04$ cm.  A bolt and a nut will fit together if the
hole in the nut is larger than the diameter of the
bolt, but the difference is no larger than $.05$ cm.
If a nut and a bolt are selected independently and 
randomly from the two boxes, what is the probability 
that they will fit together? 
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VIII.  Let $X$ and $Y$ be independent random variables with 
moment-generating functions 
$$m_X(t) = m_Y(t) = e^{t^2 + 3t}$$
\item{A)} (10) What is the moment generating function of $Z = 3X + 2Y - 4$?
\item{B)} (10) What is the distribution of $Z$?
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IX.  Twenty students in a probability and statistics class take
a final examination independently of one another.  The number
of minutes each student requires to complete the 
exam is a random variable with an exponential distribution with mean $120$ minutes. 
The students all start work at 8:30am. 
\item{A)} (10) What is the probability that at least one student
out of the 20 will complete the exam by 10:00 am?
\item{B)} (10) What is the distribution of the total time
taken by all of the students to complete the exam?
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\noindent
X.  Suppose $Y_1, Y_2$ have joint density
$$f(y_1,y_2) = \cases{24 y_1y_2 & if $y_1 \ge 0$, $y_1 + y_2 \le 1$ \cr
                           0 & otherwise\cr}$$
\item{A)}  (10)  Are $Y_1,Y_2$ independent?  
\item{B)}  (10)  What is $V(8Y_1 - 2Y_2)$?
\item{C)}  (10)  Use the method of distribution functions to find
the density function for $U = Y_1 + Y_2$.
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\noindent
{\it Extra Credit}  (10) One form of the Law of Large Numbers states that 
if $X_1,\ldots,X_n$ are independent and identically distributed random
variables for which $E(X_i) = \mu$ and $V(X_i) = \sigma^2$ exist, then for any
$\epsilon > 0$,
$$\lim_{n\to\infty} P(|\overline{X} - \mu| < \epsilon) = 1.$$
Prove this statement without assuming anything else about the
distribution of the $X_i$.   
\bye

