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\centerline{Mathematics 375 -- Probability and Statistics 1}
\centerline{Information on Exam 1}
\centerline{October 1, 2009}
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\noindent
{\it General Information}
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As announced in the course syllabus, the first midterm exam will
be given next week, either in class or in the evening on Thursday, October 8.  
The exam will cover the material we have discussed since the beginning of 
the semester, up to and including the material on binomial and geometric
discrete random variables from class on September 29.
(This is the same as the material from Problem Sets 1 - 4 and 
Discussions 1 - 2).
In more detail, the topics to be covered are:
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\item{1)}  Descriptive statistics such as the mean, standard
deviation, and frequency histograms for numerical data, the ``empirical rule''
\item{2)}  Discrete sample spaces and counting techniques
for sample points -- the $m\cdot n$ rule, permutations, 
binomial and multinomial coefficients 
\item{3)}  The ``Sample Point Method'' for probabilities
\item{4)}  Conditional probability and independence of events
\item{5)}  The Law of Total Probability and Bayes' Rule -- {\it know
proofs of these} and how to apply them.  
\item{6)}  The ``Event Composition Method'' for probabilities
\item{7)}  Discrete random variables, probability mass
functions, expected value, variance -- {\it know the proof of the 
equation $V(Y) = E(Y^2) - (E(Y))^2$ as well as how to apply it.}
\item{8)}  Binomial random variables -- {\it know the
proof of the formula $E(Y) = np$ for a binomial random variable
based on $n$ trials with success probability $p$}, and the 
formula for variance of binomial random variables.
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\noindent
{\it Other Groundrules}
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You may prepare {\it one side of a $3\times 5$ inch index card}
with formulas and any other information you want to include, 
bring it to the exam and consult it at any time. Calculators allowed.
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\noindent
{\it Review Session}
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I will be happy to run a pre-exam review session.  If we do the exam
in the evening, we could take either Tuesday or Thursday's class meeting 
for an optional review.  We would have regular class meetings the other 
two days next week, though.

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\noindent
{\it Suggested Review Problems}
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From the text:  1.22, 1.24, 1.31, 2.143, 2.144, 2.145, 2.146, 2.147 (Note: the ``full house''
would have to consist of either the pair of aces and three kings, or a pair of kings and 
three aces.  You should assume that {\it only} the original five cards have been dealt before
the discard and the deal of the two additional cards.), 2.148,
2.149, 2.150, 2.153, 2.155, 2.163, 3.15, 3.33, 3.53, 3.56, 3.61, 3.180
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\noindent
{\it Sample Exam Questions}
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\noindent
{\it The first five come from the first midterm exam in the Fall 2003
offering of MATH 375.}
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\noindent
I. A manufacturer of electronic components tests the lifetimes
of a certain type of battery and finds the following data:
$$123, 116, 122, 110, 125, 126, 111, 118, 117, 120$$
(lifetimes in hours).
How many of the sample points are within one standard deviation of 
the sample mean?  Is there reason to believe the lifetime
of this type of battery is not normally distributed from this small sample?
Explain.
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\noindent
II.  In a regional spelling bee, the 10 finalists consist
of 5 girls and 5 boys.  Assume that all the finalists are equally 
proficient spellers and that the outcome of the contest is random.
What is the probability that 4 of the top 5 finishers will be female?
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\noindent
III.  
\item{A)}  State and prove the Law of Total Probability.
\item{B)}  The Podunk City police department plans a crackdown
on speeders by placing radar traps at four different locations
$L_1,L_2,L_3,L_4$.  The probability that each of the 
traps is manned at any one time is $.4, .3, .2, .3$ respectively
(and the police really mean business -- if a trap is manned every speeder 
who passes it will get a ticket).  Speeders have probabilities 
of passing the four locations of $.2, .1, .5, .2$ respectively, 
and no one passes more than one.  What is the 
probability that a given speeder will actually receive a ticket?
\item{C)}  In the situation of part B, given 
that a speeder received a ticket, what
is the probability that he passed location $L_2$?
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\noindent
IV.  Let $A,B$ be events for which $P(A) = .2$, $P(B) = .3$ and
$P(A\cap B) = .06$.  Are $\overline{A}$ and $\overline{B}$ independent
events?
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\noindent
V.  An allergist knows that $30\%$ of all people are 
allergic to the pollen of burdock weed.  The allergist sees
20 patients in all on one day.  
\item{A)}  What is the probability mass function for
the random variable $Y = $ number of patients among the 20
who are allergic to burdock pollen?  Explain the assumptions 
you are making to derive your solution.
\item{B)}  What are the expected value and variance for the 
$Y$ from part A?
\item{C)}  What is the probability that from
4 to 7 (inclusive) of the 20 patients she sees are allergic to 
burdock pollen?  
\item{D)}   What is the probability that the first patient
the doctor sees who is allergic to burdock pollen 
will be the last patient seen?  Explain the assumptions 
you are making to derive your solution.
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\noindent
VI.  {\it Extra Credit-Type Question (New)} The Yankees won 10 of the 18 regular season games 
they played against the Red Sox in 2009.  Assume that the Red Sox and Yankees
make it to the ALCS (a big assumption of course), and that the Yankees
have a $10/18$ chance of winning each game they play against
the Sox (another {\it big} assumption at this point -- recall that the Sox
won the first $8$ games and the Yankees won the next $10$!).  Let $Y$ be 
the random variable that gives the number of the game in which
the Yankees clinch the AL pennant (if they do), and $0$ if the 
Red Sox pull out a win.  What are the probability mass function, 
the expected value, and the variance of $Y$?
\bye

