Information Pertaining to Exam 2
Basics
It'll be a open book and notes exam; you can use any printed/written materials that you bring with you (but you cannot share materials with other students).
You can use a calculator and/or a computer (but not your phone).
I'll leave room on the exam for you to work the problems, so you won't need to bring paper
or an exam book. All you'll need is a pencil, and a calculator or computer (in addition to a mastery of
the material).
The exam will consist of 6 problems having a total of 9 parts. These parts will be equally weighted, and your overall score for the exam will be the sum of your best
8 scores from the 9 parts.
Click
here
to see the instructions for your exam.
What to Study
Even though I list important things from each of the pertinent chapters below, I think it'll be best to mainly focus on studying the problems that I list near the bottom of this web page.
By solving problems and studying examples, you should get a good understanding of which parts of the text are the most important for the exam I'll give you.
Chapter 4
- Definition of cdf on p. 149, and be aware of the 4 properties given on the upper portion of p. 149-150
- Definition of pmf on pp. 158
- Definition of expected value on p. 165
- Theorem 4.2 on p. 173
- bold-faced result given right above Example 4.23 on p. 174
- Definition of standard deviation and variance on p. 179
- Theorem 4.3 on p. 181
- Theorem 4.5 on p. 182
Chapter 5 (In addition to the things listed below, focus on
when it's appropriate to use the various distributions. For example, when dealing with iid
Beroulli trials, a binomial dist'n pertains to the number of successes in a fixed number of trials, a geometric dist'n pertains to the number of trials to get the 1st
success, and a negative binomial dist'n pertains to the number of trials to get the rth success (r >= 2). Also, a hypergeometric dist'n
deals with the number of objects of one type that occur when a random subset of objects is drawn (without replacement) from a collection of two types of objects.
And one use of a Poisson dist'n is to deal with the number of events associated with a Poisson process that occur in a fixed period of time.)
- basics of Bernoulli r.v's as given on pp. 195-196
(recall that a Bernoulli r.v. is a binomial r.v. with n = 1, and so you can obtain the mean and variance by using Table 3 on p. 610)
- basics of binomial r.v's as given on p. 197 and p. 202
- know how binomial random variables relate
to iid Bernoulli trials (number of successes in a fixed number of trials))
- recall that I'll give you a copy of Table 3 on p. 610 of the text ... so no need to memorize formulas for pmf, range, mean, and variance
- Definition on p. 210 and mean and variance results near the bottom of p. 210
- Theorem 5.2 on p. 214 (know how to use Poisson pmf if just the mean is given, and also if
a rate parameter and a length of time (or distance, or something else) is given)
- Definition on p. 223 and mean and variance results on p. 223 (know how geometric random variables relate to iid
Bernoulli trials (number of trials needed to get the first success))
- Definition on p. 225 and mean and variance results on p. 226 (know how negative binomial random variables relate to iid
Bernoulli trials (number of trials needed to get r successes (r > 1)))
- basics of hypergeometric r.v's as given on pp. 227-228 (I don't like the book's terminology and notation ---
they don't have to be used for the number of defectives. Also, in the version given in the course notes, the possible values don't have to
be {0, 1, ..., n}.)
Chapter 6 (This chapter will be emphasized much more than any of the other chapters covered by the 2nd midterm exam.)
- Definition on pp. 243-244
- properties (a) through (e) on p. 244
- "cdf method" (book calls it the method of distribution functions) illustrated in Example 6.3 on pp. 253-254
(Comments: I hate the notation used in the book for this example. In the solution, I'd start off with
FY(y) = P(Y <= y)
and then use
y, FY(y), & fY(y)
everywhere the book has (respectively) t, G(t), & g(t).)
- method of transformations given in Theorem 6.1 on p. 255
- Definition on p. 259
- Theorem 6.3 on p. 262
- Corollary on p. 263 and important special case in bold right before Example 6.10 on p. 264
- Definition on p. 264 and bold-faced results right before Example 6.11 on p. 265
Chapter 7
- basics of uniform r.v's as given on pp. 279-281
- Know how to transform (using inverse cdf) a uniform (0, 1) r.v., U, to obtain a function of U which has a specified distribution.
(This isn't covered in Ch. 7, but in the class notes I inserted pertinent material from elsewhere after I covered the basics of uniform r.v's.)
- Definition 7.2 on p. 286
- bold-faced result 4 lines below Fig. 7.5 on p. 287
- know how to use Tables 1 and 2 from the appendix of the text to obtain probabilities associated with normal random variables (both standard normal
r.v's and normal r.v's which aren't standard normal)
- Definition 7.3 on p. 291
- Lemma 7.1 on p. 292
(important implications of this is given in the class notes on the last 4 lines from of p. 7-12 & the very top of p. 7-13)
- connection between exponential random variables and Poisson processes as described on middle portion of p. 299
- cdf of exponential r. variable on p. 299
- pdf of exponential r. variable on p. 299
- mean and variance of exponential r. variable on p. 300
- memoryless property on p. 302
- connection between gamma random variables and Poisson processes as described on p. 306 (also see p. 309)
- gamma and Erlang pdfs on pp. 306-307
- basics of gamma function on pp. 306-307
- bold-faced results on middle portion of p. 308
You don't have to worry about the Poisson approximation of binomial dist'n probabilities for this exam, and also the normal approximation of the binomial distribution
won't be covered.
Some good problems/examples to go over are:
- part (a) of Problem 4 from HW 4
- Problem 1 from HW 5
- parts (b) and (c) of Problem 2 from HW 5
- part (b) of Problem 1 from HW 6
- Problem 2 from HW 6
- part (a) of Problem 3 from HW 6
- Problem 1 from HW 7
- parts (a) and (e) of Problem 2 from HW 7
- parts (a) and (b) of Problem 1 from HW 8
- parts (a) and (c) of Problem 2 from HW 8
- Problem 4 from HW 8
- the example on p. 7-5 of the course notes
Here is a link to the 2nd exam I gave my class during the Fall of 2019.
A link to the solutions to this exam can be found below the link to this web page.