Information Pertaining to the Final Exam
Basics
The official exam period is 7:30-10:15 PM on Wednesday, Dec. 8.
You are expected to take the exam during the official time slot.
Exceptions to this policy will rarely be made.
(Note: The official time slot for the exam may be changed.
For example, if there are too many class cancelations due to bad
weather, or for any other reason, the Provost may alter the exam schedule.
So, to be safe, you may want to plan to remain in the area through Dec. 15,
which is on GMU's calendar as an official make-up date for exams.)
The exam is an open books and open notes exam.
You can use whatever printed or written material that you bring with you
to the exam. You cannot share books or notes during the exam.
You can use a calculator and/or computer during the exam if you wish to.
(Some may wish to use software such as Maple or Mathematica.)
You can connect to the internet to use Wolfram Alpha, a digital copy of the text, or the course web site, but using it for any other
purpose will need prior approval from me. You cannot use the internet to communicate in any way with
another party. Also, cell phones should be kept out of your hands while you're taking the exam.
Extra Office Hours
Closer to exam time, I might announce some additional office hours here.
Description of the Exam
The exam will be 7 problems having a total of 11 parts (so you'll have an average of 15 minutes per part, and some of the
parts should take a lot less than 15 minutes). Each part will be worth 10 points, and I'll count your best 10 of 11 scores from the 11 parts.
Each of the first 6 problems will have only one part. But the last problem will have 5 parts, all
pertaining to a single joint distibution of two random variables. For it you may be requested to do things like
obtain a marginal density, obtain a conditional density, obtain a conditional expectation, obtain the density or cdf of a function of the two random variables
(e.g., a sum, a difference, a product, or a ratio), obtain a covariance or a correlation, or a few other
things. (Note: If I request a conditional expectation, you might have to first obtain a conditional density, and to do that you may first have to obtain a
marginal density.)
For the other parts of the exam, here are some good hints:
- you'll be requested to give a pdf or cdf of a maximum or minimum of two random variables
(which are not iid);
- you'll be given some of the probabilities of a joint pmf, along with some information about the joint pmf (e.g., a conditional mean), and will be asked to obtain
the missing probabilities based on the information given;
- you'll be requested to obtain one or more expected values in cases for which the tactics of expressing a random variable as a sum of simpler random variables
(possibly 0/1 "indicator" random variables) or using conditional expectations may be useful (for extra practice, try to obtain the answer of 147/110 for Problem 7.23
on p. 380 of the text using both of these tactics);
- you'll be requested to approximate a probability using a result based on the central limit theorem;
- you'll be requested to obtain an mgf, or possibly use an mgf to obtain an expected value (although in this case you'd also be free to obtain the
desired expectation some other way if you wish to do so).
Keep in mind that I've stressed several ways to check your work for certain types of problems (e.g., if you obtain a pdf make sure that it integrates to 1,
and make sure a cdf goes from 0 to 1 over the support of the random variable). I encourage you to use the full
exam period, checking your work and perhaps solving some parts of the exam in more than one way (but if you arrive at a different answer using an alternative approach,
be sure to indicate which answer you want me to base your score on).
What to Study
While some basic concepts from the first five chapters (that were covered on the two midterm exams) may be useful,
the final exam will focus on Ch. 6 through Ch. 8 of the text, and the vast majority of the points will be for
Ch. 6 and Ch. 7 type problems.
My advice is to first quickly look over the highlights from each chapter that I list below, paying extra attention to the few items in bold font.
(Hopefully a lot of these things will seem very familiar.) Then turn your attention to the list of homework problems given at the bottom of this web page.
If you can get comfortable with the problems and examples indicated below, then you ought to be able to do well on the final exam. (Some of the exam problems are rather similar,
but even a bit easier, than the homework problems listed. (If you're short on time, it may be best to just focus on the list of homework problems found below, as well as the examples from the class notes that I have referred to below, and not worry about looking at everything from the book that I list here.))
Chapter 6
- formulas for obtaining marginal pdfs from a joint pdf, p. 241 (3rd line from bottom) & p. 242 (top line);
Example on p. 6-5 of the class notes, Example on p. 6-6 of the class notes
- obtaining probabilities using a joint pdf, Example 1d, p. 242; Example on p. 6-4 of the class notes
- using "cdf method" to obtain pdf of a function of two r. v's from a joint pdf, Example 1f, pp. 244-245; Example on
p. 6-7 of the class notes; Example on p. 6-9 of the class notes; Example on p. 6-12 of the class notes
- formula for conditional pdf, p. 270;
Example 5a, p. 271; Example on p. 6-16 of the class notes
- obtaining cdf and pdf of sample minimum and sample maximum, pages 6-18 and 6-19 of the class notes; Example on pages
6-19 and 6-20 of the class notes
- the method emphasized in Sec. 6.7 won't be needed for the final exam (although you could
use it for one of the parts (but I don't think it's the best way to approach that part (with the "cdf method" being perhaps the best way
to address the part)))
- parts (b), (c), (d), (e), and (f) of Problem 6.9 on p. 292
- Problem 6.19 on p. 293
- Problem 6.22 on p. 293 (and additionally obtain the covariance of X and Y from this joint pdf (a Ch. 7 task))
- Problem 6.52 on p. 295
Chapter 7
- Examples on pages 7-2 and 7-3 of the class notes (note there is more than one way to get the expected value of a sum of r. v's)
- Example 2e through Example 2j, pp. 307-310
- Example of p. 7-6 of the class notes
- Example of p. 7-8 of the class notes (but don't worry about the part that continues on p. 7-9)
- alternate formula for Cov(X,Y), top portion of p. 329 (below Definition); Example near middle of p. 7-10 of the class notes
- variance of a sum of r. v's, (4.1) on p. 330;
Example 4b, p. 332; Example on p. 7-12 of the class notes
- correlation of 2 r. v's, bottom of p. 7-14 of the class notes; Example at bottom of p. 7-15 of class notes
- conditional expectation, pp. 337-338;
Example 5b, p. 338; Example near middle of p. 7-17 of the class notes
- computing expectations by conditioning, p. 315;
Example 5d, pp. 340-341; three Examples on pages 7-18 and 7-19 of the class notes
- mgfs, pp. 360-365;
1st two Examples on p. 7-24 of the class notes;
two Examples on p. 7-25 of the class notes
- mgfs of sums of independent r. v's (and identifying the distributions of the sums), bottom of p. 364;
Example 7f,
Example 7g, and
Example 7h, pp. 366-367; three Examples on p. 7-28 of class notes
- part (a) of Problem 7.9 on p. 378
- Problem 7.13 on p. 379
- Problem 7.22 on p. 380
- Problem 7.30 on p. 380 (can be solved in several ways)
- parts (a) and (b) of Problem 7.50 on p. 382
- Problem 7.52 on p. 382
Chapter 8
- Although the various inequalities and laws of large numbers are important results, they don't necessarily lead to
good exam problems. So for this chapter you can just get by with knowing how to obtain approximate probabilities using the
central limit theorem. (See the first two examples on page 8-6 of the class notes (but don't worry about the
last example on p. 8-6).)
20 good homework and quiz problems/parts to review
- Problem 5, HW 8 (note that the normal approximation of a binomial distribution probability is a special case of using the CLT approximation)
- part (a) of Problem 8, HW 8
- part (a) of Problem 3, HW 9
- part (b) of Problem 3, HW 9
- Problem 1, HW10
- Problem 3, HW10
- part (a) of Problem 4, HW10
- Problem 5, HW10
- part (a) of Problem 1, HW12
- part (b) of Problem 1, HW12
- Problem 2, HW12
- Problem 4, HW12
- Question 1 of Quiz 8
- Question 2 of Quiz 8
- Question 3 of Quiz 8
- Bonus Question 9
- Question 1 of Quiz 10
- Question 2 of Quiz 10
- Question 3 of Quiz 10
- Question 3 of Quiz 11