\magnification=\magstep1
\centerline{Mathematics 375 -- Probability and Statistics I}
\centerline{Midterm Exam 1 -- October 8, 2009}
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{\it Directions:}  
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Do all work in the blue exam booklet.  There
are 100 possible points and 10 Extra Credit points (please turn 
over for the Extra Credit problem).
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\noindent
I.  To test the effectiveness of a seal for air leaks in automobile tires, 
after the seal was installed in a tire, a needle was inserted into the tire 
and air pressure was increased until leakage was observed.  The pressures 
(in lb per square inch) where leakage first occurred on 10 trials were:
$$93, 87, 91, 90, 92, 88, 95, 91, 90, 96$$
\item{A)} (10) Construct a relative frequency histogram for this
data on the interval $[86.5,96.5]$, subdividing into 5 equal ``bins''
$$(86.5,88.5],(88.5,90.5],(90.5,92.5],(92.5,94.5],(94.5,96.5].$$  
\item{B)} (15)  How many of the data points are within 1 standard
deviation of the sample mean?
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\noindent
II.  Sailors on a ship send signals by arranging
(exactly) 8 colored flags along a rope line.
\item{A)} (5) How many different signals can they send if
they have flags of 8 different colors?
\item{B)} (10) How many different signals can they send if
they have 4 red, 2 green, and 2 blue flags?  (There are no
differences between the flags of the same color.)
\item{C)} (10) In the situation of part B, if a random 
arrangement of flags is constructed, what is the probability that
at least 5 other flags appear between the two blue flags?
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\noindent
III.  Let $A_1,A_2,A_3$ be events in a sample space $S$.
Assume that $S = A_1\cup A_2 \cup A_3$, where $A_i \cap A_j = \emptyset$
if $i\ne j$.  Let $P(A_1) = .35$, $P(A_2) = .4$, $P(A_3) = .25$.
Finally, let $B$ be another event with $P(B|A_1) = .1$,
$P(B|A_2) = .2$ and $P(B|A_3) = .05$.  
\item{A)} (10) State the Law of Total Probability and use it to 
compute $P(B)$.  
\item{B)} (5) Are $B$ and $A_2$ independent events?  Why or why not?
\item{C)} (10) What is $P(A_1 \cup A_2|B)$?  Give the name(s) of any
general rule(s) that you are using.
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\noindent
IV.  
\item{A)}  (10)  Let $Y$ have a binomial distribution with 
parameters $n = $ number of trials and $p = $ success probability on 
each trial.  Show that $E(Y) = np$.  
\item{B)}  (15)  An army regiment has 20 different squads: 16 with 25
soldiers each, 3 with 100 soldiers each, and one with 300 soldiers,
for a total of 1000 soldiers.  Select a soldier at random out of
the 1000 and let the random variable $X$ equal the size of the squad
to which that soldier belongs.  Find the probability mass function,
the expected value, and the variance of $X$.   
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{\it Extra Credit}  (10)
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Let $X$ be a discrete random variable which takes as values all 
natural numbers $k \ge 1$.  Assuming that $E(X)$ exists, show that
$$E(X) = \sum_{k = 1}^\infty P(X \ge k).$$  
(Also, why do we need to add the hypothesis that $E(X)$ exists 
in a case like this?)
\bye
