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\centerline{Mathematics 375 -- Probability and Statistics I}
\centerline{Midterm Exam 2 -- November 19, 2003}
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{\it Directions:} Do all work in the blue exam booklet.  There
are 100 possible points.   You may leave values expressible in terms 
of factorials, binomial coefficients, etc. in those forms with no penalty.
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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 wait-person 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 all IDs show the student's actual 
age!) 
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III.  A continuous random variable $Y$ has probability density function 
$$f(y) = \cases{ {c\over y^4} & if $1 \le y \le 2$\cr
                      0 & otherwise\cr}$$
\item{A)} (5)  What is the value of the constant $c$?
\item{B)} (10)  Find the cumulative distribution function for $Y$.
\item{C)} (15)  What is the variance of $Y$?
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III.  (20) The life in years of a a certain type of electrical switch
has an exponential distribution with mean $2$ years.  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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IV.  (15) The weights of a population of miniature poodles are
normally distributed with mean $\mu = 8$kg and standard
deviation $\sigma = .9$kg.  What is the probability that
a randomly selected poodle will have weight between 
$7.3$kg and $9.1$kg?
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V.  
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