review for final, STAT 346

Information Pertaining to the Final Exam

Basics

The official exam period is 1:30-4:15 PM on Monday, December 11. You are expected to take the exam during the official time slot. Exceptions to this policy will rarely be made, and may involve getting approval from the dean's office.

The final 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. (With a computer, you can only go to the course Blackboard site, an electronic version of the text book, and Wofram Alpha (or use a software package such as Mathematica or Maple that does things similar to Wolfram Alpha). During the exam period, you may not use a phone until you turn in your exam paper to me.

Description of the Exam

The exam will be 6 problems having a total of 11 parts. Each part will be worth 10 points, and I'll count your best 10 of 11 scores.
One problem will have 6 parts. For it, you'll be given a joint pdf and asked to obtain (a) a marginal pdf, (b) a conditional pdf, (c) a conditional expectation, (d) the pdf of a function of X and Y, (e) the covariance of X and Y, and (f) the variance of a linear combination of X and Y. The other 5 problems will focus on these topics:

approximating a probability using the central limit theorem (CLT),
obtaining a probability using a linear function of independent normal random variables,
using a mfg to obtain an expectation,
obtaining the cdf or pdf of a minimum or maximum of two random variables that aren't iid (so instead of plugging into a formula, you should use the cdf method),
obtaining an expected value by expressing a random variable of interest as a sum of simple random variables (perhaps random variables that only take the values 0 and 1).

I strongly suggest that you focus your studying on making sure you know how to handle the types of problems indicated in the above description. (Consider this to be a very precise description of the exam. My guess is that seldom in your time at GMU will you be given such a narrowly focused study guide for a final exam.)

What to Study

The most important things are given below in bold font, and the very most important things are given below in red bold font. Don't get bogged down with the details of the definitions. I think it'll be best to focus on understanding the important problems and examples listed towards the bottom of this web page. (For most of the things listed below, I hope that you'll be able to take a glance, nod your head in recognition that it seems familiar, and quickly move on to the next item.)

Chapter 8
(Although discrete random variables are important too, to make the exam easier to prepare for, I'll limit coverage of this chapter to the parts pertaining to continuous random variables.)

Definition 8.3 on p. 333 (alternatively, see the 6th line below the Joint Desities heading on p. 8-8 of the course notes for a more user-friendly version)
(8.5) & (8.6) on p. 335
Definition 8.4 on p. 335
bold-faced results on bottom half of p. 336 (pertaining to expected values)
Theorem 8.2 on p. 341
Corollary on p. 341
1st 3 lines of Sec. 8.2 on p. 348
Theorem 8.3 on p. 349
Theorem 8.5 on p. 350
Theorem 8.6 on p. 350-351
Theorem 8.7 on p. 352
top line of p. 356
(8.21) and (8.22) on p. 366
(8.24) on p. 368
(8.25) on p. 369
the "cdf method" for finding the pdf of a function of two random variables (strongly emphasized in class but not text, and so study the examples on pages 8-13, 8-15, 8-23, and the bottom half of 8-33 of the course notes)

Chapter 9

know how to obtain the pdf of the sample minimum and sample maximum of two continuous random variables (know this for the case of iid random variables, with independent but not identically distributed random variables, and also if the random variables are not independent)

Chapter 10

Theorem 10.1 on p. 429
Corollary on p. 430
Definition on p. 439
Theorem 10.4 on p. 440
(10.4), (10.5), and (10.6) on p. 440
(10.9), (10.10), (10.11), and (10.12) on p. 444
Lemma 10.1 on p. 446
Definition on p. 451
Theorem 10.5 on p. 451

Chapter 11

Theorem 11.1 on p. 484
Theorem 11.6 on p. 495
Theorem 11.7 and bold-faced result following theorem on p. 496
Theorem 11.12 on p. 520, Remark 11.2, and Remark 11.3 on p. 521, and know how to obtain approximate probabilities involving sums and sample means of iid random variables using the CLT (see bottom of p. 11-22 of the course notes)

a few examples from the class notes to try to understand well

example on bottom half of p. 8-11
example on p. 8-31
example on p. 10-7
in the example on pages 10-14 and 10-15 know how to obtain the marginal pdf of X and the value of E(XY)
using the mgf derived in the example on p. 11-4 to obtain the value of E(X)
using the mgf derived in the example on p. 11-5 to obtain the value of E(X)
example on p. 11-14
example on p. 11-23
example on p. 11-24

good homework problems to review

Problem 1, HW 9 (even though this is based on Ch. 7, some of the skills used in obtaining the answer are useful for solving CLT problems of Ch. 11 (try solving it using p. 11-22 of the notes))
parts (a) and (b) of Problem 7, HW 9
parts (a) and (b) of Problem 1, HW 10
Problem 2, HW 11 (and also obtain the conditional expectation given in the parenthetical comment)
Problem 4, HW 11 (I recommend that you also obtain the cdf of the minimum of X and Y.)
Problem 1, HW 12
Problem 2, HW 12
Problem 3, HW 12
Problem 4, HW 12
Problem 7 of HW 12 from my 2010 class
Problem 5 of HW 13 from my 2010 class

Link to four more nice problems (with the 1st and 3rd problems perhaps being the best ones to do if you're short on time to prepare for the exam), and a link to solutions for parts 2(g), 3(b), & 4(a).