Math 361 X1

Exam 2 Study Guide

Friday, April 4, 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. (There is no need for a normal table in this exam, as the normal distribution will not come up.)

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.

Exam content

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. 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.

• 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 a single formula. For example, I would not ask for a joint distribution if it were a simple matter of multiplying two given individual distributions.

• If you get entangled in lengthy computations, you are likely on the wrong track. All exam problems have a simple solution that takes no more than a couple of lines and does not require messy computations. Brute force approaches are almost always counterproductive, they waste valuable time, and in most cases will not lead to the right answer anyway. You will not be given extra time to complete a brute force approach if a simpler alternative method is available. You should leave final answers in raw form, such as (171/4) + (27/4)², or in terms of binomial coefficients or factorials; trying to work out numerical answers will not earn you additional credit, and makes the grading more difficult.

• Most problems on the exam will be of a familiar type. Since the number of different types of problems on the exam material is quite limited, most exam problems will be mathematically similar to problems you have encountered before, such as class examples or homework problems. Try to spot such similarities, but also be on the lookout for small but significant differences; for example, don't try to blindly apply the method for the birthday problem, just because a problem involves the word birthday. The best preparation for the exam (in addition to memorizing relevant definitions, rules, and formulas) is to study the HW problems and class examples.

• Write-up of solutions. As usual, you need to show your work; an answer alone is insufficient. A formal definition of Omega and A as sets is in most cases unnecessary, but in combinatorial problems, you need to indicate how you arrived at each component in your answer. E.g., if the answer is (8.6.4.2.2)/(8)₅, you should say how you arrived at of the factors in the numerator 8, 6, etc., and the numerator (8)₅.

Exam content

• Combinatorial Probabilities: Some material can be found in Section 2.5 and Appendix 1 of the Pitman text, but for the most part you should base your preparations on class notes (and the class summaries on the course web page), and the supplementary handouts that were distributed in class. This material is largely problem-oriented, and the best preparation consists in working examples and problems like those in class, in the HW assignments, or on the supplementary handout. The main topics are combinations (unordered samples); permutations (ordered samples); and box/ball type problems.

• 3.1: Random variables; Introduction: For this material, and 3.2 (Expectations), you can follow closely the excellent presentation in the Pitman text. You can skip p.141 (bottom) - 143. You should know the basic definitions and formulas (distribution, joint distribution, marginal distributions, independence - see p. 145, 151, 154, and the formula summary on p. 248), and know how to work with these concepts. You can skip the parts on conditional distributions and symmetry, from p. 150 through the end of the section.
  Most problems in this section fall into the following two distinct types:
  • (I) Given a description of the random variable(s), find its distribution (or joint distribution). Problems of this type boil down to classical probability computations of the type done in Chapters 1 and 2. In order to do these problems, you need to be familiar with the methods introduced in those chapters. If a problem asks for a distribution, be sure to include the range.

  • (II) Given a distribution of a r.v, (or a joint distribution of two r.v.'s), compute probabilities involving these r.v.'s. These problems can be solved in a systematic way, by writing the probability asked for as a sum over individual probabilities P(X=x) (or P(x,y)=P(X=x,Y=y)), where the summation condition mirrors the condition on X and Y in the probability sought, then using the known information about the values (or ranges) for x (or x and y) to reduce this abstract summation to a concrete sum over specific values of x and y, and finally substituting the (given) probabilities (P(X=x) or P(x,y)) into this formula.
  You should also be familiar with the basic concepts and definitions of this section, such as distribution, joint distribution, marginal distribution, and independence, and you should be prepared for problems involving these concepts.

• 3.2: Expectation of random variables. Here again, the Pitman text serves as an excellent guide. Read this section, except for the part "Expectation and prediction" on p. 178 - 179. There are three main ways to compute the expectation of a r.v.
  (i) A direct computation from the distribution, using the formula (definition) of E(X), and the values of P(X=x) given in the distribution. If the distribution is given or easy to compute, this is usually the best approach.
  (ii) Use properties of E(X) (such as the addition formula, or the product formula for independent r.v.'s) (see p. 181 for a summary of these properties).
  (iii) The indicator method. This method works only for a particular type of r.v. X, namely when X is the number of events A_i that occur, but when the method does apply, it often provides a relatively easy way to compute expectations that would otherwise be (near) impossible to compute. The key to applying the method is to identify (define) the events A_i.

• 3.4/3.5 Discrete distributions: In 3.4, skip Examples 3 (moments), and 5 (Collector's problem). From 3.5, only the definition of the Poisson distribution is required. You can skip the parts on skew-normal approximation and random scatter.
  Important distributions: From the distribution summary on p. 476, you should know the formulas for the probabilities P(X=k) and the expectations of the uniform, binomial, geometric (on {1,2,...}), and Poisson distributions. The formulas for variances are less important and harder to memorize, and the only distribution for which you should know the variance is the binomial distribution.

• Chapter 3 Summary, p. 248 - 249: You need to know the formulas and definitions on the first page (p. 248), and the properties of expectation summarized on p. 181.

• Chapters 1 and 2: Many problems involving random variables boil down to probability computations of the type that have come up in Chapters 1 and 2, such as S/F probabilities, birthday type problems, sampling problems (e.g., lottery), and you should (still) know how to do such problems. If you are a bit rusty, be sure to review this material.

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Last modified Sat 29 Mar 2003 01:09:29 PM CST