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\centerline{Mathematics 375 -- Probability and Statistics 1}
\centerline{Midterm Exam 2a}
\centerline{November 16, 2005}
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{\it Directions}
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Do all work in the blue exam booklet.
There are 100 regular points and 10 Extra Credit points.
If you know general formulas for means, variances, etc. you can
use them without comment (i.e. you don't need to rederive them in 
you solution). 
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\noindent
I. (10) In a shipment of 14 television sets, 5 have defective ``on''
switches and will not power up.  Out of the shipment, 7 sets are
selected at random.   What is the probability that at most 3 will 
not power up?
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\noindent
II.  
\item{A)} (10) The number of defects $Y$ in any 1-foot segment
of a magnetic tape is a Poisson random variable with 
mean $5$.  What is the probability that a 1-foot segment 
contains exactly 4 defects?  
\item{B)} (15) Show, by any applicable method, that
if number of defects in one 1-foot segment is $Y_1$ and 
the number in a second 1-foot segment (not overlapping the first)
is $Y_2$, then the total number of defects, $Y = Y_1 + Y_2$, has a Poisson
distribution with mean 10.  (Assume defects
in the two segments are independent.)
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\noindent
III.  In a biomedical research experiment, the survival time $Y$
(in weeks) of an animal exposed to a certain fatal disease is a random 
variable with a gamma distribution with $\alpha = 3$ and $\beta = 2$.
\item{A)} (5) What is the mean survival time of a randomly selected 
exposed animal?
\item{B)} (10) Use Tchebysheff's Theorem to estimate
the probability that a single exposed animal will 
survive longer than 29 weeks.
\item{C)} (15) If 6 exposed animals are chosen at random, use
your answer from part B to estimate the probability
at least 2 of them survive longer than 29 weeks.
\item{D)} (15) The cost of treating an exposed animal is 
$C = 5 + .2 Y^2$.  Find the mean and variance of $C$.
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\noindent
IV. 
\item{A)} (10) The lifetime of a certain type of small motor is 
{\it uniformly} distributed on the interval $[0,30]$.  What
is the probability that the motor will last more than 10 years?
\item{B)}  (10) Now, assume the lifetime of the motors is 
{\it normally} distributed with mean 15 years and standard deviation 
4 years.  The manufacturer replaces free all motors that fail while 
under warrantee.  But to avoid going out of business because of the 
cost of doing the free replacements, the manufacturer must set the 
warrantee so that they will have to replace 
no more than 3\% of the motors that fail.  What should the length 
of the warrantee be?
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\noindent
{\it Extra Credit} (10) 
Let $Z$ be a standard normal random variable.  What is the 
moment-generating function for $Y = 8Z - 4.3$?  What is the 
distribution of $Y$?
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