# Math 408: Actuarial Statistics I, Spring 2008 Professor A.J. Hildebrand

## Final Exam (scores are in!)

• Final Exam Solutions.
• Statistics. The maximal total score was 75 points, the median was 60, and the 80th/60th/40th/20th percentiles were 68/62/57/49. For the score breakdown per problem see the solutions.
• Online scores: The online score display shows your score on the final and your course grade. An explanation of the score display can be found under this link.
• Special Math 408 Office Hours. If you want to see your final exam, stop by my office, 241 Illini Hall, Monday, May 5, 1 pm - 2 pm, or Tuesday, May 6, 4 pm - 5 pm.

Have a great break and the best of luck to those
taking the May Actuarial Exam!

• Midterm Exam 1: Friday, March 7, 9 am - 9:50 am, 66 Library.
• Midterm Exam 2: Wednesday, April 23, 9 am - 9:50 am, Room 1320 DCL. The exam will cover the material since the first midterm.
• Final Exam: Friday, May 2, 7 pm - 10 pm, 103 Mumford Hall.
• James Scholar Honors Credit Projects. A list of projects, and details on what you are expected to do in order to receive James Scholar Honors Credit for this class. (This is for James Scholars only. If you are not a James Scholar, this doesn't apply to you.)

## Class Diary

• Monday, 4/28:
• Friday, 4/25:
• Lecture. More combinatorial problems. Ordered versus unordered counting. Birthday problem variants.
• Wednesday, 4/23: Midterm Exam II.
• Monday, 4/21:
• Friday, 4/18:
• Lecture. Sums and linear combinations of independent normal r.v.'s. This is the final topic on the Central Limit Theorem handout, and wraps up Chapter 5. (I will not cover the remaining sections.)
• Read. Section 5.3 covers this material, but it is somewhat hidden in between other topics such as the chi-square distribution, which I will not cover here. The most relevant example is 5.3-3. In the exercises, the relevant problems are 5.3:10-15. (There will be no additional graded homework assignment, but I will make solutions available before the end of the semester.)
• Wednesday, 4/16:
• Lecture. More examples on normal approximation, and informal description/explanation of the underlying mathematical phenomenon.
• Monday, 4/14:
• Friday, 4/11:
• Lecture. More on the normal distribution.
• Read. Section 5.2 through p. 274 (Example 5.2-7). You can skip the final part (Theorem 5.2-2 and Examples 5.2-8 and 5.2-9), since it requires the chi-square distribution which we have not covered.
• Wednesday, 4/9:
• Monday, 4/7:
• Lecture. Section 4.5. Several independent random variables, random samples, the max/min trick.
• Read. Section 4.5. The most important part of this section are the problems involving the maximum or minimum of several independent random variables, such as the last three problems in this week's hw assignment.
• Quiz. The quiz tomorrow will be on joint continuous distributions and on the maximum/minimum trick from today's class.
• Homework: Homework Assignment 10, due Friday, 4/11
• Friday, 4/4:
• Lecture. Tips and strategies for word problems. More examples from Problem Set 4.
• Wednesday, 4/2:
• Monday, 3/31:
• Friday, 3/28:
• Lecture. Continuous joint distributions, continued. Uniform joint densities.
• Wednesday, 3/26:
• Lecture. Started the discussion of continuous joint distributions, with some problems illustrating probability computations via double integrals. See Sections 4.1, 4.2, 4.3 of Hogg/Tanis, where the continuous case is covered alongside the discrete one. This is one of the most important topics in this course, but also one that most students find difficult (because of the double integrals involved), so I plan to spend at least another week on this material.
• Handout. Joint distributions, continuous case
• Monday, 3/24:
• Friday, 3/14:
• Wednesday, 3/12:
• Lecture. Discrete joint distributions, continued. Independence, covariance and correlation.
• Monday, 3/10:
• Lecture. Discrete joint distributions, continued. Conditional p.m.f.'s, conditional expectations and variances. Independence.
• Quiz. The quiz tomorrow will be on discrete joint distributions, i.e., the material on the Joint Discrete Distributions handout from last Wednesday, and illustrated by the examples in Wednesday's and today's lectures.
• Friday, 3/7: Midterm Exam.
• Wednesday, 3/5:
• Lecture. Started Chapter 4 (multivariate distributions). Discrete joint distributions, and their properties. Representation as distribution matrix. Marginal distributions. Computations with (discrete) joint distributions: probabilities involving X and Y, expectations, variances, conditional probabilities, expectations, variances.
• Handout. Joint distributions, discrete case
• Homework: HW Assignment 7, due Wednesday, next week (3/12). (Note the deadline!)
• Monday, 3/3:
• Lecture. Change of variables in random variables. The distribution function technique. (This is the last bit of new material I'm covering from Chapter 3 for now.)
• Read. Section 3.5 covers this material. The method I illustrated in class, and which I recommend to use for all change of variables problems, is called in the book the "distribution function technique, and illustrated by Examples 3.5-1 and 3.5-2 (you need not know the names or formulas for the special distributions (gamma, loggamma, Cauchy) used in these examples). (Note that he book also discusses another method involving an explicit formula for g(y) (e.g., in Example 3.5-3 and 3.5-7), but that method is fraught with pitfalls, and not applicable in all situations, so I don't recommend it. Every change-of-variable problem can be handled with the distribution function technique, so there is no need to learn yet another method.)

You can skip over the two theorems (3.5-1 and 3.5-2) in this section.

• Friday, 2/29:
• Wednesday, 2/27:
• Lecture. More on the exponential distribution. Typical applications. The "no memory" of the exponential distribution. Actuarial problems.
• Read. Section 3.3. This section covers the uniform and exponential distributions. Note that the exponential distribution here is discussed in the context of the Poisson process, a more complicated notion, which I didn't cover in class and which are not expected to know. You can skip over the examples/problems that involve the Poisson process (e.g., Ex. 3.3.-5 or Problem 3.3-8); most of the problems, though, require no knowledge of the Poisson process.
• Monday, 2/25:
• Lecture. Continuous random variables, continued. Exponential and uniform distributions.
• Quiz. The quiz tomorrow will be on the formulas from the last week's handout on continuous distributions.
• Homework: Homework Assignment 6, due Friday, 2/29
• Friday, 2/22:
• Lecture. Continuous random variables, continued: Review of basic concepts, differences to the discrete case, some common errors and pitfalls. An example from an actuarial exam.
• Wednesday, 2/20:
• Lecture. I started Chapter 3 (continuous distributions). General concepts and formulas: probability density function (p.d.f.) f(x), cumulative distribution function (c.d.f.) F(x), median and percentiles.
• Read. Section 3.2. (Section 3.1 is largely a motivational section. It's an interesting read, and it is not needed for the remainder of the chapter, or for the actuarial exam.)
• Handout: Continuous Random Variables. Summary of concepts and formulas on continuous random variables.
• Monday, 2/18:
• Lecture. Actuarial exam problems on discrete random variables.
• Quiz. The quiz tomorrow will be on the formulas from the last week's handout on the formulas for the "Big Three" discrete distributions: binomial, geometric, and Poisson.
• Handout. Actuarial Exam Problem Set 2
Solutions to Problem Set 2
• Homework: Homework Assignment 5, due Friday, 2/22
Note on 2.6-21 and 2.6-23: If you got 342 dollars (or so) as answer for 2.6-21 and 2.681 as answer for 2.6-23(a), you have the correct answers. If the answers in the back of the book are different (older editions of the text have the wrong answers), just ignore them. (In the case of 2.6-21, \$342 is the expected penalty (\$400 per bumped passenger), while the answer in the book, \$598, is the expected total payment (\$700 per bumped passenger). Mathematically the arguments are exactly the same, so don't loose sweat over this.)
• Friday, 2/15:
• Lecture. The Poisson distribution. This is the last of the three major discrete distributions; it typically arises in connection with rare events like snowstorms or hurricanes.
• Read. The Poisson distribution is covered in Section 2.6 of Hogg/Tanis, but it is lumped together with another, more complicated concept, the Poisson process. You don't need to know the Poisson process for this class (and it is not needed for the actuarial exam either), so you can skip over the parts of this section dealing with the Poisson process (in particular, the definition on p. 113 and Example 5).

What you do need to know are the formulas summarized on the Discrete Random Variables handout: p.m.f., expectation, variance of a Poisson distribution, and the exponential series formula. As with the binomial distribution, when working problems involving the Poisson distribution, you should not use tables for this distribution (as the book does in Example 2.6-1), but work solely with the formulas for the p.m.f.

• Wednesday, 2/13:
• Lecture. Independent success/failure trials, continued. Examples involving the geometric distribution.
• Read. The geometric distribution can be found in Section 2.5, Be aware that this section (like most of the other sections in Chapter 2) is a jumble of several different topics (moment-generating functions, negative binomial distribution, geometric distribution), and I have been proceeding differently in class. I covered the moment-generating function earlier, and I won't cover the negative binomial distribution. The only formulas you need are those on the two handouts on Discrete Random Variables distributed in class.
• Monday, 2/11:
• Lecture. Independent success/failure trials ("Bernoulli trials". The binomial distribution.
• Handout. Discrete Random Variables, II
• Read. The material is covered in Section 2.4 of the book. Two caveats on the treatment in the book:
• In several of the examples (e.g., 2.4-8) the book refers to a table (Table II) for computing binomial probabilities. Do not follow this practice, but rather compute the probabilities directly. In actuarial exams (and our in-class exams and quizzes), you don't have a binomial table available, so you have to learn to do without one.
• The last example (2.4-11) requires the hypergeometric distribution which we have not covered yet. Ignore that example.
• Quiz. The quiz tomorrow will be on the formulas from the first handout on Discrete Random Variables (from last week). This is the handout that covers the formulas for general distributions. Today's handout, which covers formulas for special distributions (binomial, etc.), will not be on the quiz.
• Homework: HW Assignment 4, due Friday, 2/15.
• Friday, 2/8:
• Lecture. Expectation, variance, standard deviation, moment-generating functions, continued. More examples. Interpretations expectation and variance. Some common mistakes.
• Wednesday, 2/6:
• Lecture. Recap: Discrete Random variables, p.m.f.'s (probability mass functions). Type I vs. Type II problems. Probability computations with p.m.f.'s. Expectations.
• Monday, 2/4:
• Lecture. Started Chapter 2 on Discrete Random Variables. Motivation and informal definition of a random variable, examples. Probability mass function (p.m.f) (discrete density function): definition and general properties.
• Handout. Discrete Random Variables, I
• Read. Sections 2.1/2.2; focus on the following: 2.1 through Example 2.1-3, and 2.2, through Example 2.2-4. The later examples in those sections involve combinatorial probabilities (Section 1.3), a topic that I have not yet covered. Also, in discussing discrete distributions I will proceed in a different order than the book does in Chapter 2, focusing first on the abstract set-up and general formulas and properties (as given on the first page of the above handout), and then discussing particular discrete distributions (as given on the second page of handout). The book spreads out the development of the general theory (p.m.f., expectation, variance, moment general functions etc.) over several sections, and intersperses the discussion of particular distributions into the general build-up of the theory.
• Quiz. The quiz tomorrow will be on the material from last week: conditional probabilities, independence, and the total probability and Bayes' rules. Discrete random variables will not on this week's quiz.
• Homework: HW Assignment 3, due Friday, 2/8.
• Friday, 2/1:
• Lecture. Discussed the airline problem from HW 1, and the disease test/abused child test problems from HW 2. Both illustrate situations where "numbers can trick you", and in which the results are the opposite of what one might expect, but can be explained by a rigorous mathematical analysis using Bayes' Rule and the total probability rule. Handed out a set of Actuarial Exam practice problems, and worked out one problem (#15, the blood pressure/irregular heartbeat problem).
• Handout. Actuarial Exam Practice Problem Set 1
Solutions to Problem Set 1
• Wednesday, 1/30:
• Monday, 1/28:
• Friday, 1/25:
• Lecture. Sections 1.4/1.5: conditional probabilities and independence, continued.
• Wednesday, 1/23:
• Lecture. Sections 1.4/1.5: conditional probabilities, independence. (Section 1.3 will be deferred till the end of Chapter 1.)
• Handout: Independence and conditional probability.
• Homework: HW Assignment 1, due Friday, 1/25.
• Read. Sections 1.4/1.5. Focus on the abstract set-theoretic examples (e.g., 1.4-2 or 1.4-3), and examples like 1.4-11 involving insurance/actuarial applications. You can skip over examples involving card problems or selections with/without replacement (e.g., Ex. 1.4-6 or 1.4-7).
• Friday, 1/18:
• Lecture. Examples illustrating the use of probability rules, Venn diagrams, and translating probabilistic language into set-theoretic language.
• Quiz. There will be a short quiz at the beginning of Tuesday's discussion section on the material from the two handouts distributed on Wednesday and covered in today's class (set-theoretic computations using probability rules, Venn diagrams, and translations between set-theoretic and event language).
• Wednesday, 1/16:
• Lecture. Section 1.2: Axiomatic definition of probability. Sample space S, outcomes (elements of S), events A (subsets of S), probability function P(A). Rules for probability functions. Kolmogorov axioms, derived rules. Venn diagrams and area rule of thumb.
• Handouts.
• Read. Section 1.2 (focus on the examples; you can skip over the proofs).
• Monday, 1/14:
• Lecture. First-day Handout. Overview of this course. Probability versus statistics. Basic concepts: Probabilistic experiment, outcomes, events, outcome space.
• Read. Prologue, Section 1.1 (especially the examples), Appendix A1 (review of set theory).

## Announcements

• [3/22/08] James Scholar Honors Credit Projects. The above link provides a list of projects, and details on what you are expected to do, in order to receive James Scholar Honors Credit for this class. (Note that the projects are intended for students in the James Scholar program who have submitted an Honors Credit Learning Agreement (HCLA) for this class. If you are not a James Scholar, this doesn't apply to you.)
• [12/12/07] Text for this course: The text will be Hogg/Tanis, "Probability and Statistical Inference" (7th edition). It is (or will be) available at all local college bookstores. At a list price of around \$130, it does not come cheap. However, you can get so-called "international editions", which have the same content, but a different (soft) cover, for less than half that price from various online outlets. I got my copy from TextbooksRus.com for \$52.48, including shipping. (Disclaimer: I have no relation with this outfit other than as a satisfied customer. Other deals can be found at campusbooks.com.)
• [11/13/07] Full Sections. This is a course that tends to fill up fast. Unfortunately, there is nothing I can do about this. I have no control over available slots or registration caps, and I cannot "squeeze" someone in who isn't able to register because of a full section. Please do not email me with requests to sign you up for the course; as much as I'd like to accommodate everyone, there is nothing I can do if the registration limit is reached.

All I can advise you is to wait and see if spaces open up. This frequently happens during the first few days of the semester due to dropouts. In fact, in past years there have always been enough drops so that everyone who wanted to be in this class could (eventually) be accommodated. As a backup, you might want to sign up for a section of Math 461, the regular undergraduate probability class.

Note that in order to be able to register for this course, there must be spots available in both the lecture and one of the discussion sections. If there are spots available in the lecture, but both discussion sections are full, you will not be able to register.