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\centerline{Mathematics 375 -- Probability Theory}
\centerline{Review Sheet, Final Exam}
\centerline{December 1, 2011}
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\noindent
{\it General Information}
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The final examination for this course will be given at 
3:00 pm on Friday, December 16 in our regular class room,
Swords 359.  The exam will be roughly twice the length of 
one of the two midterms and I will let you continue
working on it as long as you want (even after the formal end
of the exam period at 5:30pm, if necessary).
As was true on the midterms, I will let you bring a sheet
of information for your use on the exam -- for this exam, 
you can place anything you want on {\it one side of an 
8.5 $\times$ 11 inch piece of paper}.   I will provide
copies of whatever tables from the text might be needed
to work some of the problems.  
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\noindent
{\it Topics to be Covered}
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This will be a comprehensive exam -- all topics we have
discussed this semester are ``fair game.''  This includes:
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\item{1)}  Descriptive statistics such as the mean, 
standard deviation, frequency histograms, etc.  The
``empirical rule''.
\item{2)}  Discrete sample spaces and counting techniques
(especially in connection with the ``sample point method''
for the probability of an event):  the $m\times n$ rule, permutations, 
binomial and multinomial coefficients.
\item{3)}  The ``event composition method'' for 
probabilities
\item{4)}  Conditional probabilities, independence of
events, Bayes' Rule, the Law of Total Probability
\item{5)}  Discrete random variables: probability 
distribution functions, cumulative distribution
functions, expected values and variances of functions
of a discrete random variable, moment generating 
functions.  Know the situations leading to 
binomial, geometric, Poisson, and hypergeometric random
variables and how to apply them.    
\item{6)}  Continuous random variables: probability 
distribution functions, cumulative distribution
functions, expected values and variances of functions
of a continuous random variable, moment generating 
functions.  Know the situations leading to uniform,
exponential, gamma, beta, and normal random
variables and how to apply them.   
\item{7)} Tchebysheff's Theorem.
\item{8)}  Multivariate probability distributions:
joint densities, marginal and conditional densities,
expected values in this setting, conditions for 
independence, the covariance and 
the general formula for the variance of a linear
combination of random variables.    
\item{9)} Using moment generating functions and the 
uniqueness theorem to determine the distribution of 
a random variable.
\item{10)}  The method of distribution functions to
determine the distribution of 
a random variable.
\item{11)}  Know the statement and proof of the 
Central Limit Theorem.
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\noindent
{\it Suggestions on How to Study}
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Start by reading the above list of topics carefully.
If there are terms there that are unfamiliar or for which
you cannot give the precise definition, start by reviewing
those topics.  Review the class notes.  {\it Everything on 
the final will be closely related to something we have discussed at
some point this semester.}   Also look back over your graded
problem sets and exams.  If there are problems that you 
did not get the first time around, try them again now, consulting
the solutions on reserve in the Science Library as necessary. 
Then go through the suggested problems from the 
review sheets.  If you have worked these out previously,
it is not necessary to do them all again.  But try a representative
sample ``from scratch'' -- don't just look over your old
solutions.  Practice thinking through the logic of how
the solution is derived again.
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\noindent
{\it Suggested Practice/Review Problems}
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Look at the problems from the two previous
review sheets for the topics from Chapters 1 - 5.
From Chapter 5/105 (do all the computations needed to 
find $V(Y_1 - Y_2)$ using the formula of Theorem 5.12), 147,149,151,161.  
Chapter 6/93,95,107,111.
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\noindent
{\it Review Session}
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I will be happy to run a review session for the final
exam.  We can discuss a time in class on Friday, December 9.
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\noindent
{\it Sample Exam Questions}
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\noindent
{\it Note:  This was the final exam given in the section of 
this class taught in 2009.  Solutions for these questions are
available on the course homepage for that section -- go to my 
personal homepage, follow links Course Homepages/Previous Course Home Pages,
and look at the page for MATH 375 from Fall 2009.}
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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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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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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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\noindent
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

