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\centerline{Mathematics 375 -- Probability and Statistics 1}
\centerline{Midterm Exam 2b}
\centerline{November 18, 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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I. (10) In a group of 12 chickens, 5 are infected with 
the H5N1 ``bird flu'' virus and the others are not.
If 7 chickens are selected randomly from the group,
what is the probability 2 or more have the virus?
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II.  
\item{A)} (15) Assume that
$Y_1$ has a Poisson distribution with mean $\lambda_1$
and $Y_2$ has Poisson distribution with mean $\lambda_2$.
Also assume that $Y_1 = y_1$ and $Y_2 = y_2$ are independent 
events for all $y_1 \ge 0$ and $y_2\ge 0$.  Show, by any applicable method, 
that $Y = Y_1 + Y_2$ also has a Poisson distribution, with mean 
$\lambda = \lambda_1 + \lambda_2$.  
\item{B)} (10) The number of defects $Y$ in any 2-foot segment
of a magnetic tape is a Poisson random variable with 
mean $3$.  Use part A to find the probability that a 4-foot segment 
contains exactly 8 defects, assuming the necessary independence.
(You can use part A even if you didn't completely see how to do
that part.)
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III. 
\item{A)} (10) The lifetime of a certain type of battery is 
{\it uniformly} distributed on the interval $[0,25]$ (years).  What
is the probability that the battery will last more than 10 years?
\item{B)}  (10) Now, assume the lifetime of the batteries is 
{\it normally} distributed with mean 10 years and standard deviation 
4 years.  The manufacturer replaces free all batteries 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 5\% of the batteries that fail.  What should the length 
of the warrantee be?
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IV.  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 {\it gamma distribution} with $\alpha = 4$ and $\beta = 3$.
\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 30 weeks.
\item{C)} (15) If 6 exposed animals are chosen at random, use
your answer from part B to estimate the probability
no more than 2 of them survive longer than 30 weeks.
\item{D)} (15) The cost of treating an exposed animal is 
$C = 4 + .25 Y^2$.  Find the mean and variance of $C$.
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{\it Extra Credit} (10) 
Let $Z$ be a standard normal random variable.  What is the 
moment-generating function for $Y = 3Z+5$?  What is the 
distribution of $Y$?
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