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\centerline{Mathematics 375 -- Probability and Statistics I}
\centerline{Review Sheet, Final Exam}
\centerline{December 5, 2005}
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\noindent
{\it General Information}
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The final examination for this course will be given at 
8:30 a.m. on Saturday, December 17 in our regular class room,
Smith Hall 210.
The exam will be roughly twice the length of one of the three
midterms, but you will have the full three hour period from
8:30 am to 11:30 am to work on it if you need that much time.
As was true on the midterms, I will let you bring a sheet
of information for your use on the exam -- for this exam, 
you can place anything you want on {\it one side of an 
8.5 $\times$ 11 inch piece of paper}.   I will provide
copies of whatever tables from the text might be needed
to work some of the problems.  
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\noindent
{\it Topics to be Covered}
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\item{1)}  Descriptive statistics such as the mean, 
standard deviation, frequency histograms, etc.  The
``empirical rule''.
\item{2)}  Discrete sample spaces and counting techniques
(especially in connection with the ``sample point method''
for the probability of an event):  the $m\times n$ rule, permutations, 
binomial and multinomial coefficients.
\item{3)}  The ``event composition method'' for 
probabilities
\item{4)}  Conditional probabilities, independence of
events, Bayes' Rule, the Law of Total Probability
\item{5)}  Discrete random variables: probability 
distribution functions, cumulative distribution
functions, expected values and variances of functions
of a discrete random variable, moment generating 
functions.  Know the situations leading to 
binomial, geometric, Poisson, and hypergeometric random
variables and how to apply them.    
\item{6)}  Continuous random variables: probability 
distribution functions, cumulative distribution
functions, expected values and variances of functions
of a continuous random variable, moment generating 
functions.  Know the situations leading to uniform,
exponential, gamma, beta, and normal random
variables and how to apply them.   
\item{7)} Tchebysheff's Theorem.
\item{8)}  Multivariate probability distributions:
joint densities, marginal and conditional densities,
expected values in this setting, conditions for 
independence, the covariance and 
the general formula for the variance of a linear
combination of random variables.    
\item{9)} Using moment generating functions and the 
uniqueness theorem to determine the distribution of 
a random variable.
\item{10)}  Know the statement  of the
Central Limit Theorem.
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\noindent
{\it Suggestions on How to Study}
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Start by reading the above list of topics carefully.
If there are terms there that are unfamiliar or for which
you cannot give the precise definition, {\it learn the definitions
now; memorize them if you have to!}  You simply cannot answer
a question about this material if you do not know what the 
terms mean.  Review the class notes.  {\it Everything on 
the final will be similar to something we have discussed at
some point this semester.}  Also look back over your graded
problem sets and exams.  If there are problems that you 
did not get the first time around, try them again now, consulting
the solutions on reserve in the Science Library as necessary. 
Then go through the suggested problems from the 
review sheets.  If you have worked these out previously,
it is not necessary to do them all again.  But try a representative
sample ``from scratch'' -- don't just look over your old
solutions and nod your head if it looks familiar.  
Practice thinking through the logic of how
the solution is derived again.
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\noindent
{\it Suggested Review Problems}
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Look at the problems from the two previous
review sheets for the topics from Chapters 1 - 4.
From Chapter 5/11,27,28,57,58,59,61,84,96,120,121,122,123,125,133,134
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{\it Review Session}
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I will be happy to run a review session for the final
exam during study week.  We can discuss a time in class
on Monday, December 5.
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\noindent
{\it Sample Exam}
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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)} 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)} 
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)} 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.  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)}  What is the probability that a randomly 
selected seed will germinate?
\item{B)}  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)}  Find the mean and variance of $Y$.
\item{B)}  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)}  What is the probability that a single mint has weight
$ < 20.4$ grams?  
\item{B)}  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)}  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. 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)} 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)} Determine the value of $k$.
\item{C)} Find $E(Y)$ and $V(Y)$.
\item{D)} 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)} What is $P(Y_2 > Y_1)$? ({\it Hint:} Draw a picture!)
\item{B)} Are $Y_1$ and $Y_2$ independent?  Why or why not?
\item{C)} Determine $V(3Y_1 + 2Y_2)$.
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\noindent
{\it Extra Credit}
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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?
\bye

