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\centerline{Mathematics 375 -- Probability and Statistics 1}
\centerline{Final Examination -- December 18, 2003}
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\noindent
{\it Directions}
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You have {\it until 5:30 p.m.\/} to work on this exam.
Do problems I - IX in the blue exam booklet.  There are 200 possible 
regular points and 20 Extra Credit points.  Notes:
\item{1)}  If an answer can be expressed in terms of binomial
coefficients, the gamma function, factorials, etc., it is 
{\it preferable} to leave it in that form.
\item{2)} It is OK (even suggested!) to use general formulas we have
derived, etc. rather than working out probability integrals, 
etc. from scratch.
\item{3)}  Even if you 
don't see how to derive a complete solution to a problem, don't leave
it blank.  Partial credit will be given for relevant definitions, 
ideas that could lead to solutions, and so forth.  
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\noindent
I.  A large number of observations of a certain continuous
random variable $Y$ were taken,
and a typical subset of $n=50$ of them were ordered to produce
the data set below:
$$\matrix{
0.10&0.23&0.25&0.26&0.26&0.28&0.31&0.39&0.42&0.44\cr
0.45&0.48&0.53&0.54&0.55&0.66&0.69&0.70&0.72&0.73\cr
0.73&0.74&0.74&0.75&0.77&0.78&0.81&0.95&0.97&1.02\cr
1.03&1.05&1.07&1.13&1.13&1.14&1.20&1.22&1.27&1.28\cr
1.30&1.34&1.34&1.37&1.39&1.40&1.43&1.43&1.46&1.48\cr}$$
\item{A)} (10) Construct relative frequency histograms for 
this data using 5 bins on the interval $[0,1.5]$, and 
then 10 bins on the interval $[0,1.5]$.
\item{B)} (10) 
Two models are proposed for the distribution of $Y$:
\itemitem{1.} a {\it uniform} distribution on 
$[0,1.5]$, or 
\itemitem{2.} a distribution described by the 
probability density function (pdf)
$$f(y) = \cases{{8y\over 9} & if $0\le y \le 3/2$\cr
                    0 & otherwise\cr}$$
Find the expected values and variances of the two
proposed model distributions.  
\item{C)} (10) The actual sample mean is
$\overline{y} = .8542$ and the sample standard deviation
is $s \doteq .4033$.  How well does the ``empirical rule''
match the distribution of $Y$?  

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\noindent
II.  (20) A bin contains three components from supplier $A$, four
from supplier $B$, and five from supplier $C$.  If four
of the components are selected randomly for testing, 
what is the probability that each supplier will have at least
one component tested?

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\noindent
III.  Carrot seeds from grower $A$ have an $85\%$ germination
rate, while seeds from grower $B$ have a $75\%$ germination and 
those from grower $C$ have a $90\%$ germination rate.
rate.  A seed packaging company buys $30\%$ of its seeds
from grower $A$, $40\%$ from grower $B$, and $30\%$ from grower
$C$, then mixes them thoroughly in making up packets for sale.
\item{A)} (10) What is the probability that a randomly 
selected seed will germinate?
\item{B)} (10) Given that a seed did not germinate,
what is the probability that it came from grower $B$?

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\noindent
IV.  A permutation of $A = \{1,2,3,4,5\}$ is a one-to-one, onto mapping
from this set to itself.  For instance, the $f$ defined by 
$$f(1) = 2, f(2) = 4, f(3) = 3, f(4) = 1, f(5) = 5$$
is one such permutation.  Consider a permutation selected at
random, and let $Y$ be the {\it number of elements of the set
$A$ that are mapped to themselves}.  For instance, if the 
$f$ above was selected, then the value of $Y$ is  $2$ 
(since $f(3) = 3$ and $f(5) = 5$).  
For this problem,  you are given the information that the 
moment-generating function of $Y$ is 
$$m(t) = {44\over 120} + {45\over 120}e^t + {20\over 120} e^{2t} + 
{10\over 120}e^{3t} + {1\over 120} e^{5t}$$
\item{A)}  (10) Find the mean and variance of $Y$.
\item{B)}  (10) What is the probability that a randomly selected
permutation maps {\it at least one element of $A$ to itself}?

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\noindent
V.
A candy maker produces thin chocolate mints (yum!) that have a label
weight of 20.4 grams.  The manufacturing process is subject to 
some randomness, though, and the actual weights of the mints
are normally distributed with mean $\mu = 20.8$ and $\sigma = 0.3$.
\item{A)}  (10) What is the probability that a single mint has weight
$ < 20.4$ grams?  
\item{B)}  (10) If mints are selected independently and at random
from the production line, what is the probability that at least
10 trials will be necessary to find one that has weight $< 20.4$ grams?
\item{C)}  (10) If 40 mints are selected independently and at
random from the production line, what is the probability 
that 3 or fewer will have weight $< 20.4$ grams?

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\noindent
VI. (10) 900 students are given an exam.  The average score is 
$\overline{y} = 83$ and the sample variance is $v = 36$.
At least how many students must have scored between $71$ and $95$?

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\noindent
VII.  Telephone calls arrive at a switchboard.  
\medskip
\item{A)} (10) The number of calls within a 10 minute interval
has a Poisson distribution with mean $\lambda = 5$.
What is the probability that more than $10$ calls arrive
in a 10 minute interval?
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\noindent 
Starting from any
instant, the elapsed time $Y$ until the 5th call arrives is 
a random variable with a pdf of the form
$$f(y) = \cases{k y^4 e^{-2y} & if $y > 0$\cr
                      0 & otherwise\cr}$$
\item{B)} (10) Determine the value of $k$.
\item{C)} (10) Find $E(Y)$ and $V(Y)$.
\item{D)} (10) The elapsed time $Y$ until the 5th call and the 
elapsed time $Z$ between the 5th and 10th calls are independent,
and both have the same distribution.
Find the distribution of the random variable $Y + Z$ by using the 
appropriate moment-generating functions.

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\noindent
VIII.
Let $Y_1$ and $Y_2$ be random variables with joint pdf 
$$f(y_1,y_2) = \cases{ 3y_1^2 y_2 & if $-1 \le y_1 \le 1, 0 \le y_2 \le 1$\cr
                            0 & otherwise\cr}$$
\item{A)} (10) What is $P(Y_2 > Y_1)$? ({\it Hint:} Draw a picture!)
\item{B)} (10) Are $Y_1$ and $Y_2$ independent?  Why or why not?
\item{C)} (10) Determine $V(3Y_1 + 2Y_2)$.
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\noindent
{\it Extra Credit} (20)
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Let $Y$ be a continuous random variable whose cumulative
distribution function $F(y)$ satisfies $F(y) = 0$ for all $y \le 0$.
Assume that $Y$ has the ``memoryless'' property:
$$P(Y > a + b | Y > a) = P(Y > b)$$
 for all $a,b > 0$.
Show that $g(y) = 1 - F(y)$ satisfies 
$$g(y + y') = g(y)g(y')$$
for all $y, y' > 0$.  What type of distribution does $Y$ have?
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\centerline{\it Have a peaceful and joyous holiday season!}
\bye