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\centerline{Mathematics 375 -- Probability and Statistics 1}
\centerline{Information on Exam 2}
\centerline{November 9, 2005}
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\noindent
{\it General Information}
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The second midterm exam will be given
either Wednesday evening November 16, or in 
class on Friday, November 18.  The exam will
cover the material we have discussed since the first exam, 
up to and including the material on Tchebysheff's Theorem
and moment generating functions we discussed on Friday, November 4.
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\noindent
{\it Format and Groundrules}
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This will be a {\it closed book} exam, but 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.  
I will provide copies of the tables for Poisson and normal
probabilities from the text.  Calculators allowed.
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\noindent
{\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 Monday, November 14 would be best if we
do the exam on Wednesday, but
other times might be possible too. 
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\noindent
{\it Suggested Review Problems}
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From the text: Chapter 3/157, 158, 160, 173,
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. (15) Ten college students, exactly four of whom are not of legal age, 
go out to a bar and all order alcoholic beverages.  The waitperson taking 
their order selects 5 students at random from the group 
to ``card'' and must refuse to serve any one who is under legal age.  
What is the probability that exactly 2 students will be refused service?
(Assume, unrealistically perhaps, that each ID shows that student's actual 
age!) 
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\noindent
II.  The fraction $Y$ of impurities in a tank-car lot of industrial grade
potassium perchlorate is a random variable with probability density function 
$$f(y) = \cases{ c y^2(1-y)^2 & if $0 < y < 1$\cr
                      0 & otherwise\cr}$$
\item{A)} (10)  What is the value of the constant $c$?
\item{B)} (10)  What is the variance of $Y$?
\item{C)} (10)  Find the cumulative distribution function for $Y$.
\item{D)} (10) Find $P(.1 \le Y \le .25)$.  
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\noindent
III.  The life in years of a certain type of electrical switch
has an exponential distribution with mean $2$ years. 
\item{A)} (15)  What is the probability that a single switch will
fail during the first year after it is installed?
\item{B)}  (5) If 
100 of these switches are installed in different, independently
operating systems, what is the probability that at most 30 fail
during the first year?  (You need {\it not} obtain a single decimal 
approximation to this number; a formula for computing it will suffice.) 
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\noindent
IV.  The weights of a population of miniature poodles are
normally distributed with mean $\mu = 8$kg and standard
deviation $\sigma = .9$kg.  
\item{A)}  (15) What is the probability that
a randomly selected poodle will have weight between 
$7.3$kg and $9.1$kg?
\item{B)}  (10)  Let $Y$ be the weight of a poodle from the 
population as a random variable.  What is $E(Y^3)$?  (Note: part credit
will be given for a correct verbal description of a procedure that
could be used to derive this value even if you cannot carry out
the procedure completely.)
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\noindent
{\it Extra Credit} (10) Let $Y$ be a random variable
with probability density function 
$$f(y) = \cases{ y e^{-y} & if $y > 0$\cr
                   0 & otherwise\cr}$$
Find the moment-generating function $m_Y(t)$, assuming $t < 1$.
\bye

