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\centerline{Mathematics 375 -- Probability and Statistics I}
\centerline{Midterm Exam 2 -- November 19, 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 
question IV and the Extra Credit problem.)
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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)  What is the probability that a baby will have
birth weight less than $2165$ grams?
\item{B)}  (5)  What is the probability that a baby will have birth
weight between $2866.5$ grams and $3763.5$ grams?
\item{C)}  (10)  If newborn babies are selected randomly, 
how many babies would you expect to have to weigh
to find the first one that weighs between $3763.5$ grams and $4258$ grams?
\item{D)}  (5)  If we knew the mean and standard deviation, but we did not 
know the distribution of the birth weights, what could we say in general about
the probability in part A?  
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II.  
\item{A)}  (10)  Fifteen out of a batch of 
twenty identical-looking packets of white crystals contain sugar and
five contain artificial sweetener.  
Four packets are randomly selected, found to contain
sugar, and discarded.  What is the probability that if three
packets are selected from those remaining, at least one contains
sugar?
\item{B)}  (10)  The number of customers arriving at a 
coffee shop to place orders has a Poisson distribution with 
mean $20$ per hour.   Give a formula for computing the probability
that more than $40$ customers arrive in a given hour.  (It is not necessary
to obtain a decimal approximation for this number.) 
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III. 
\item{A)} (5) Let 
$$g(x) = \cases{ {1\over 2} e^{x/2} & if $x < 0$\cr
                         0 & otherwise.\cr}$$
Show that $g$ is a legal pdf.  (This is a ``reflected exponential.'')
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For the remainder of this problem let 
$$f(y) = \cases{{1\over 4} e^{y/2} & if $y < 0$\cr
                {1\over 18} y e^{-y/3} & if $y > 0$\cr}$$
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\item{B)} (10)  Show that $f(y)$ is a legal pdf for a random variable $Y$.
\item{C)} (10)  Show that $E(Y) = 2$.  (Don't work harder than necessary!)
\item{D)} (10)  Determine the moment-generating function of $Y$.  On
which interval of $t$ values is your formula valid?
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IV.  Let $Y_1,Y_2$ be jointly continuous random variables with joint density
$$f(y_1,y_2) = \cases{c y_2 & if $0 \le y_1 \le 1$ and $y_1^2 \le y_2 \le 1$\cr
                   0 & otherwise\cr}$$
\item{A)}  (10)  What is the value of $c$?
\item{B)}  (10)  What is $P( Y_1 \le Y_2 )$? 
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{\it Extra Credit.} (10)  Given two real numbers $a, b$ with $a < 0 < b$,
it is possible to find a random variable $U$ whose moment-generating function is
$$m_U(t) = {1\over (1 - at) (1 - bt)}$$
for some interval of $t$ in ${\bf R}$?
If so, give a formula for the pdf for such a random variable.  If not, say
why not.  (Hint:  Think about what happened in problem III above.)
\bye
