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\centerline{Mathematics 375 -- Probability Theory}
\centerline{Midterm Exam 2 -- November 17, 2011}
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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.  Birth weights of babies born in the US are normally distributed
with mean $\mu = 3315$ grams and standard deviation $\sigma = 575$ grams.
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\item{A)}  (5)  Find the probability that a randomly chosen baby will have
birth weight more than $3700$ grams.
\item{B)}  (10)  If 20 newborn babies are selected randomly, how
many of them would be expected to have weights between $2740$ and $3890$ grams?
\item{C)}  (10)  If we knew the mean and standard deviation, but we did not 
know the precise distribution of the birth weights, what could 
we say in general about the probability that a randomly selected baby 
has weight differing from $3315$ grams by at least $862.5$ grams?  
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II.   (15) The number of defects $Y$ in a 1-foot segment
of a magnetic tape has a Poisson distribution with 
mean $3$.  What is the probability that a 1-foot segment 
contains more than 4 defects?    
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III. (10) (From an actuarial ``P Exam'' problem) The warranty on a machine 
specifies that it will be replaced at failure or at age 4, whichever 
occurs first.  The machine's age at failure, $X$, is a random variable 
with probability density function
$$f(x) = \cases{1/5 & if $0 < x < 5$\cr
                 0  & otherwise\cr}$$
Let $Y$ be the age of the machine {\it at replacement}.  What is $E(Y)$?
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IV. Let 
$$g(x) = \cases{ {1\over 16} x^2 e^{x/2} & if $x < 0$\cr
                         0 & otherwise (that is, if $x \ge 0$).\cr}$$
\item{A)} (10) Show that $g$ is a legal pdf for a random variable $X$. 
\item{B)} (10)  Determine the moment-generating function of $X$.  On
which interval of $t$ values is your formula valid?
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V.  Let $Y_1,Y_2$ be jointly continuous random variables with joint density
$$f(y_1,y_2) = \cases{c y_1 & if $0 \le y_1 \le 1$ and $y_1^2 \le y_2 \le 1$\cr
                   0 & otherwise\cr}$$
\item{A)}  (15)  What is $P( Y_1 \le Y_2 )$?  (Note that you will need to determine $c$.)
\item{B)}  (15)  Find the marginal density $f_1(y_1)$ and use it to 
compute $V(Y_1)$.
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{\it Extra Credit.} (10)  What is $E(X)$ for the random variable $X$ from question IV?  
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