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\centerline{Mathematics 375 -- Probability and Statistics I}
\centerline{Makeup Exam 2 -- December 11, 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 more than $3890$ grams?
\item{B)}  (7.5)  What is the probability that a baby will have birth
weight between $2383.5$ grams and $4005$ grams?
\item{C)}  (7.5)  If 30 newborn babies are selected randomly, what 
is the probability that 6 of them weigh between $2383.5$ grams and $4005$ 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)  Twelve out of a batch of 
twenty identical-looking packets of white crystals contain sugar and
five contain artificial sweetener.  
Three packets are randomly selected, found to contain
sugar, and discarded.  What is the probability that if four
packets are selected from those remaining, no more than one contains
sugar?
\item{B)}  (10)  The number of customers arriving at a 
sandwich shop to place orders has a Poisson distribution with 
mean $14$ per hour.   Give a formula for computing the probability
that at least $20$ 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 
$$g(x) = \cases{ {1\over 3} e^{x/3} & if $x < 0$\cr
                         0 & otherwise.\cr}$$
Show that $g$ is a legal pdf.  (This is a ``reflected exponential.'')
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For the remainder of this problem let 
$$f(y) = \cases{{1\over 6} e^{y/3} & if $y < 0$\cr
                           & \cr
                {1\over 2} y e^{-y} & if $y > 0$\cr}$$
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\item{B)} (10)  Show that $f(y)$ is a legal pdf for a random variable $Y$.
\item{C)} (10)  Find $E(Y)$.  (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
$$f(y_1,y_2) = \cases{c y_1 & if $0 \le y_1 \le 1$ and $0\le y_2 \le y_1^2$\cr
                   0 & otherwise\cr}$$
\item{A)}  (10)  What is the value of $c$?
\item{B)}  (10)  What is $P(Y_1 \le 3/4, Y_2 \le 1/2)$? 
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{\it Extra Credit.} (10)  If $Y$ has a standard normal distribution, then
$X = e^Y$ is said to have a lognormal distribution.  Determine $E(X)$ and
$V(X)$ for $X$ with a lognormal distribution.
\bye
