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\centerline{Mathematics 375 -- Probability and Statistics I}
\centerline{Midterm Exam 1 -- October 7, 2005}
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{\it Directions:}  Do all work in the blue exam booklet.  There
are 100 possible points and 10 Extra Credit points.
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I.  To test the effectiveness of a seal designed to keep an electrical
plug airtight, a needle was inserted into the plug and air pressure
was increased until leakage was observed.  The pressures (in lb per square
inch) where leakage first occurred on 10 trials were:
$$8.1, 8.5, 8.3, 8.5, 8.2, 7.8, 8.9, 7.5, 8.3, 8.0$$
\item{A)} (5)  What is the sample mean?
\item{B)} (10)  How many of the data points are within 2 standard
deviations of the mean?
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\noindent
II.  Students on a boat send signals back to shore by arranging
(exactly) 9 colored flags on a vertical flagpole.
\item{A)} (5) How many different signals can they send if
they have flags of 9 different colors?
\item{B)} (10) How many different signals can they send if
they have 4 green and 5 red 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
all the green flags appear {\it consecutively}?
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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 $P(A_1) = .3$, $P(A_2) = .2$, $P(A_3) = .5$.
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) What is $P(B)$?
\item{B)} (5) Are $B$ and $A_2$ independent events?  Why or why not?
\item{C)} (10) Find $P(A_1 \cup A_2 | B)$.
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IV.  
\item{A)}  (15)  A jar contains 7 red and 11 white balls.  Draw {\it two
balls} at random, {\it without replacing the first ball after it is drawn}.  
Let $X$ be the number of balls drawn that are red.  Thinking of $X$ as a 
discrete random variable, find its probability mass function, and compute 
its expected value and variance.
\item{B)}  (10)  Assume you have a 40\% of connecting each
time you dial a very busy customer service telephone line.
If 25 calls are made at random and independently, what is the  
probability that between 15 and 20 of them (inclusive) will get through?
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{\it Extra Credit}  (10)
Let $Y$ have a geometric distribution with success probability $p$.
Show that the expected value of $g(Y) = e^Y$ is $E(e^Y) = {pe\over 1-qe}$.  
\bye
