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\centerline{Mathematics 375 -- Probability and Statistics 1}
\centerline{Information on Exam 2}
\centerline{November 13, 2003}
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\noindent
{\it General Information}
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As announced in the course syllabus, the second midterm exam will
be given in class next {\it Friday, November 21}.  The exam will
cover the material we have discussed since the first exam, 
up to and including the material on Tchebysheff's Theorem
we discussed on Friday, November 7.
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\noindent
{\it Format}
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This will be an {\it open book, open class notes exam}.  I'm doing that to 
try to the minimize the pressure to memorize lots of stuff.  But
you should still prepare very carefully for the exam and 
understand the key concepts we have talked about.  Know
how to apply the different techniques we have studied.
If you are flipping through the book to look for {\it every} formula 
you need or if you need to find examples similar to the test questions
to get started on a problem, you {\it will not be able to finish the exam 
in 50 minutes}.
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{\it Topics}
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\noindent
The topics to be covered are:
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\item{1)}  The hypergeometric and Poisson discrete probability 
distributions -- Sections 3.7 and 3.8.
\item{2)} Moment-generating functions for discrete random variables --
Section 3.9.
\item{3)}  Continuous random variables and probability distributions,
cdfs and pdfs -- Section 4.2
\item{4)}  Expected values for continuous random variables -- Section 4.3
\item{5)}  Uniform, Normal, Gamma, and Beta distributions -- Note
that this includes the definition and properties of the Gamma and 
Beta functions.  Be prepared
for problems like those on the homework where you need to find
the mean and variance of some {\it function of} a random variable
of one of these types, and also for problems where
you might use the geometric or binomial distributions
in combination with one of these -- Sections 4.4 - 4.7
\item{6)}  Moment-generating functions for continuous random 
variables -- Section 4.9
\item{7)}  Tchebysheff's Theorem -- Section 4.10
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\noindent
{\it Review Session}
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I will be happy to run a pre-exam review session if there is interest.
The evening of Wednesday, November 19 would be best, but
other times might be possible too. 
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{\it Suggested Review Problems}
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From the text: Chapter 3/157, 158, 160, 173, 176cd, 178,
Chapter 4/67, 88, 101, 110, 128, 129, 52 and 131, 132, 133, 135, 136, 137, 142, 150, 
151
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\noindent
{\it Sample Exam}
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\noindent
Disclaimer:  The following problems represent the approximate
length and level of difficulty of the upcoming exam; the actual exam
problems may differ substantially from these.
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\noindent
Comments:  Any general formula we have studied (or that you 
find in the text!) for mean, variance, etc. can be
used without comment (i.e. you don't need to rederive it in 
you solution).
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\noindent
I.  In a shipment of 10 missiles, there are 3 defectives whose
engines will not fire.  Out of the shipment, 4 missiles are
selected at random. 
\item{A)}  What is the probability that all 4 will fire?
\item{B)}  What is the probability that at most 2 will not fire?
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\noindent
II.  In a biomedical research experiment, the survival time $Y$
in weeks of an animal exposed to a certain fatal disease is a random variable
with a gamma distribution with $\alpha = 2$ and $\beta = 6$.
\item{A)} What is the mean survival time of a randomly selected exposed animal?
\item{B)} The cost of treating an exposed animal is $C = 10 + .1 Y^2$.
Find the mean and variance of $C$.
\item{C)} If 6 exposed animals are chosen at random, what is the probability
that at least two survive for 15 or more weeks?
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\noindent
III.  The lifetime of a certain type of small motor is normally distributed
with mean 10 years and standard deviation 2 years.  The manufacturer
replaces free all motors that fail while under warrantee.  But to 
avoid going out of business because of the cost of doing the 
free replacements, the manufacturer must set the warrantee so that
no more than 3\% of the motors that fail will be covered.  What
should the length of the warrantee be?
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\noindent
IV.  
\item{A)}  Let $Y$ be a binomial random variable based on $n$ trials
with success probability $p$.  Find the moment-generating function 
of $X = aY + b$, where $a, b$ are constant.
\item{B)}  Let $Z$ be a standard normal random variable. What is the 
moment-generating function for $Y = Z^2$?
What type of random variable is $Y$?
(Hint:  You will want to assume that $1 - 2t > 0$ and use
the substitution $u = \sqrt{1 - 2t}\cdot y$ to evaluate the integral
for $E(e^{tY})$.)
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