Math 361 X1

Exam 1 Study Guide

Friday, Feb. 28, 12 - 1 pm, 260 MEB

Rules

Calculators/books/notes: Calculators will not be needed for this exam, they would be distraction and are therefore not allowed. You can and should leave answers in "raw" form - e.g., leave binomial coefficients unevaluated except in the simplest cases, or leave numerical answers, such as 6/Pi^2, unevaluated. Books or notes are not allowed, but you will be provided with a copy of the normal distribution table on p. 531 of the book (the same table as the one distributed in class).

Grading: After scoring the exam, I will set a curve by specifying cutoffs for A's, B's, etc. The cutoffs depend on the distribution of scores, but I do not try to achieve predetermined percentages of A's, B's, etc. This allows for some flexibility by giving, for instance, no F's, or more than the "proper" share of F's. Once the exam has been graded and the scores have been entered into the computer grading system, you can access your scores online, at the following web site: https://www-s.math.uiuc.edu.edu/bluestem/grades.cgi. (Instructions on accessing and interpreting these "Score reports" will be provided shortly. The raw HW scores are already in the system, and you can use this link to check your scores.)

Exam content

The exam will be on the material covered in class through Friday, 2/21. (but not the material (combinatorial probabilities) to be covered in class on Monday (2/24) and Wednesday (2/26) the week of the exam). This corresponds to Chapters 1 and 2 of the Pitman text, except for the following parts:

Empirical distributions (p. 29)
Odds (p. 6, and bottom of p. 51)
Section 1.6
Skew-normal approximation (p. 104 - 108)
Section 2.3
Section 2.5

There will be around five (plus or minus one) problems. One or two problems will be of a more theoretical nature (definitions, probability rules); the remaining problems will be word problems, comparable in difficulty and length to the homework problems. The problems are intended to test your knowledge and understanding of the various probabilistic models that were introduced in class, and they must be solved using the methods discussed in class for dealing with these models. You should model your solutions on those presented in class and in the homework solutions. You will not receive credit for an answer alone, or for a solution by a "method" that we haven't covered in class. Most students who think you can do a problem by a different method, are probably wrong and will get the wrong answer (de Mere's "method" is a good example of that), and if they happen to get the correct answer, their solution will likely not pass the "mathematical rigor" test and therefore not count.

General advice

Read the problem carefully (several times over) to make sure that you completely understand what the problem is about. Try to rephrase, in your own language, what the problem asks for. This is the most important piece of advice one can give for doing word problems. Little words like "if" can make a huge difference. There will be no trick problems on the exam, and all problems can be solved with the information given in the problem, and are stated as unambiguously as possible. In cases where a problem seems to have several different interpretations, there is only one interpretation under which the problem makes good sense as a problem, while all other interpretations are implausible or contrived. If you are not certain about the proper interpretation, go through the possible alternatives and ask yourself in each case whether this is the intended interpretation; usually, a clear winner will emerge.
How to do word problems. Getting correct numerical answers in a word problem is the least important aspect in solving the problem. Far more important, and more difficult, is to put a word problem into a rigorous mathematical framework, and to solve the problem within that framework. The nature of the problem determines which of the various models that we have discussed is appropriate. Usually there is only one model that can be used, and it is not always obvious which model that is. Experience with problems (such as the examples worked out in class or the HW problems) here is very helpful. The number of different types of problems is quite limited, and you may recognize a problem as being mathematically equivalent to a problem you have come across before.
If your solutions appears to be too simple to be true, it probably is, and chances are that you are missing a subtle, but essential, point. It is unlikely that a problem can be completely solved by simply reciting a single formula, such as the formula for binomial probabilities. Most problems will require multiple steps, and a mathematically precise set-up that involves defining all relevant events, etc.
Write-up of solutions. An essential part in solving a problem is the proper write-up, using correct mathematical notation. An answer alone, or a numerical calculation without justification will not earn any credit. Use the homework solutions and class examples as models for writing up your solutions. In particular, for word problems you may need to introduce an appropriate outcome space Omega, specify what a single outcome corresponds to in the given problem, identify the events in question as subsets of Omega, and compute the probabilities asked for within this model. For some problems (such as problems on conditional probabilities), the nature of Omega is immaterial and it suffices to specify the events in question as sets, along with any given probabilities. If a problem can be modelled by repeated success/failure trials, say so and specify the meaning of "trial", "success", and "failure".

Preparing for the Exam

Memorize all relevant formulas and definitions. These are conveniently listed in the Chapter summaries on p.72 - 73 and p.130-131. From these summaries, you should know all formulas except the following:
- Odds, p. 73
- Consecutive odds rations, p. 131
- Mode of binomial distribution, p. 131
- Random sampling, p. 131
For normal approximation, use the formulas from the class handout.
Review/study the solutions to class examples and HW problems. You should get to the point where you completely understand these problems and would get a perfect score if you had to do them over again. Start with class examples (which are often a bit easier), then go on to do the HW problems, and finally go on to do some examples in the book, concentrating on those examples (and exercises) recommended in the Reading Guide to the Pitman Text.
For additional practice, work some of the problems in the book. Stick to those problems that were specifically recommended in the Reading Guide. reading guides Other problems from the text may require techniques we haven't discussed in class or may be unsuitable for other reasons.
Additional resources. There is no shortage of probability books on the market. The one by Pitman does an excellent job in explaining and motivating the subject, and is closest to what we are doing in class. The Pitman book together with class notes and HW solutions are all you need to prepare adequately for the exam (and will likely keep you busy for some time ...). If you want to use other sources, this should be in addition, not in place of, the things I mentioned. I that case, my first recommendation would be ``Probability and Statistics'' by Murray Spiegel from the Schaum outlines series. This book (like all books in the Schaum series) has tons of practice problems (many of which are solved), and is especially useful as a refresher on certain topics (such as set theory). The relevant parts for this exam are Chapter 1 ("sets and probability"), and from Chapter 4 the sections on the binomial distribution, normal distribution, normal approximation, and Poisson distribution. The book is inexpensive (around $20) and should be available at local bookstores. (Note that there are two other Schaum outlines books on probability. One of these, titled "Probability" by Seymour Lipschutz, would also be suitable -- it covers essentially the "Probability" part of the above book. The other, "Probability, Random Variables, and Random processes" contains more advanced material.)

Exam topics

Theory

You should be familiar with the basic concepts (outcome space, event, ...) definitions (independence, conditional probability, ...) and rules (addition rule, complement rule, Bayes' rule, ...), and the relevant set-theoretic notation and language. All of these are conveniently collected in the chapter summary on p.72/73. You can skip the sections on "Interpretations" and "Odds", but you should know (i.e., memorize) everything else. Note that for theoretical problems (such as deriving a formula/rule from known probability rules), you should use the mathematical definitions, not the interpretations. For example, when working with conditional probabilities, use the definition P(A|B) = P(AB)/P(B), not an interpretation such as "prob. that A occurs if B has occurred".
Homework problems: HW 2, # 2 - 4, HW 3 # 2

Equally likely outcome models

Many problems can be solved using probability models in which all outcomes are equally likely. In those cases, the formula P(A)=#(A)/#(Omega) applies. In the vast majority of cases, the outcomes can be represented by tuples of numbers. Typical situations where such a model is appropriate are problems involving repeated rolls of a die; birthday-type problems; and sampling problems. While computing #(Omega) is usually easy, computing #(A), the number of outcomes in the event in question can be trickier. The best strategy for counting all tuples in a given set A is often a step-by-step approach: first, determine the number of ways to fill the first "slot" in the tuple, then multiply this number by the number of ways for filling the second slot (assuming the first slot has been filled), etc.
Ordered vs. unordered: For almost all problems, using ordered tuples as outcomes is, better than using "unordered tuples". In many cases, the latter approach does not work at all, and in those few cases where it could be used, the former method (with ordered tuples) is usually also applicable and much safer.
Homework problems: HW 1, # 1 - 4, 5*; HW 2, # 1 (variation on the birthday problem); HW 3, # 1
(Problems marked by an asterisk are at the high end of the difficulty scale. They are not representative of a typical exam problem, though there may be one such problem on the exam. If you are short of time, skip those problems. Make sure you fully understand the other problems, which are more routine, before spending time on an asterisk problem.)

Success/failure trial models

Problems that lend themselves to this type of model are those involving a sequence of independent trials with two relevant outcomes (head/tail, success/failure, six comes up/ no six comes up, etc.). For problems involving rolls of a die, this model can be used if the only thing that matters is whether or not a six comes up; otherwise, a model involving tuples as outcomes has to be used. For example, the standard birthday problem cannot be modelled by S/F trials, since the exact birthday (in the range 1 - 365) is relevant, not just whether that birthday is or is not a particular number.

General probability computations in S/F models: If an event does not fall into one of the familiar special cases (see below), you have to perform "assembly level computations": Write down explicitly all S/F sequences that form the event in question, and obtain the probability for that event by adding up the probabilities of each of these sequences.
Homework problems: HW 3, # 3

Probabilities involving the number of successes in a S/F model: The binomial distribution gives the probability for getting k successes in n trials; many problems can be expressed in terms of such probabilities.
Homework problems: HW 3, # 4 - 8;

Conditional probabilities

Word problems involving conditional probabilities should be done in the same rigorous manner as the examples worked out in class, the handouts, and the homework solutions. The key to these problems is the set-up which requires, in particular, defining and introducing notation for all relevant events and translating the given data into probabilities involving these events.
Homework problems: HW 2, # 5 - 7.

Poisson approximation

The Poisson distribution can, under certain conditions, be used to approximate the binomial distribution. It is essential that you know what these conditions are and do not apply the approximation where it is not appropriate.
Homework problems: HW 4: For some of the problems Poisson approximation is the appropriate tool. (You'll have to figure out which problems those are. HW solutions will be posted Friday, 2/23.)

Normal approximation

Use the class handout for reference. Know the formulas on this handout, if you haven't done so already. In the exam, you will get a copy of the normal table on the back of this handout, but not the front containing the formulas for normal approximation. Also, read the remarks which contain you some tips on applying these formulas. Here are some key points to keep in mind when applying normal approximation:

Formulas: There are two main formulas (see the handout), one involving the small phi function, the other involving the capital phi function. You should know both formulas, and know when to apply each of these formulas.

Computations: While phi(y) is defined by a simple formula involving the exponential function, no such formula exists for Phi(z). Values of Phi(z) are tabulated in the "Normal Table" (p. 531 in the text, handed out in class on 2/18), and you should practice using this table. If you find that a value of Phi is out of range for the table, then most likely you did something wrong, such as applying the normal approximation in a situation where it was not appropriate.

Conditions for the normal approximation: The normal approximation is accurate when n is large, p not too small or too close to 1. More specifically, p and 1-p should be large compared with 1/n. (If p is of order 1/n, Poisson approximation can be used.) Also, normal approximation should not be used for probabilities falling into the "tail ends" of the distribution. In general, the approximation is best when the parameter k (the number of successes) is in the "belly" of the distribution, which occurs near np.

Square root law, p. 100: This is more a rule of thumb than a well-defined mathematical "law". It may give you an idea of what to expect, but you cannot use this "law" when doing problems; for that, you will need to apply the normal approximation formula.

Confidence intervals: You should be familiar with the concept of confidence intervals (for the number of successes and for the success probability p), but you should not memorize probabilities associated with specific confidence intervals. However, you should be able to derive these probabilities on your own, using the normal table.

Homework problems: Most problems on HW 4 require normal approximation.

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Last modified Sun 23 Feb 2003 12:45:31 PM CST