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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 answers in terms of
of factorials, binomial coefficients, etc. if appropriate.
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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 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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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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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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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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{\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 $t < 1$.
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