A. Adhikari

TEXT READING GUIDE

Your lecture notes should be your starting point because in each lecture I will try to summarize the main ideas and use instructive examples. If you read your lecture notes thoroughly first, then the path through the text will be easier.

It goes without saying that the Exercises at the end of each section are important. This page lists what I want you to read before you try those exercises. You can skim at first but if you have trouble with the exercises then read the section carefully before trying the exercises again.

CHAPTER 1

1.1: Pages 1-5.

1.2: Pages 11-13. It is important to understand the figure on page 13.

1.3: This is the basis of everything that follows. The table on Page 19 should be imprinted on your heart and you should thoroughly understand pages 20-25. Skip Example 4 for now. Learn to recognize the Bernoulli distributions and the uniform distribution on a finite set. In this course it is a very good idea to learn to recognize the standard distributions that crop up frequently.

1.4: Try following this path: start with Example 2 on page 34. This motivates the general formula at the top of page 36. Read that formula and Example 4. Then come back to Example 3 – it's the interesting one. The multiplication rule on page 37 will come as no surprise. At this point you should be able to simply read Examples 6 and 7. Compare the general multiplication rule on page 37 with the special case when you have independence, on page 42. Then skim Examples 8 and 9. See if you can draw a tree diagram that can be used in Example 9. You should now be able to go back and do Example 4 of 1.3 without reading the solution first.

1.5: Page 47 through the discussion that ends at the top of page 51.

1.6: This obvious extension of the familiar multiplication rule leads to a lot of interesting stuff. I suggest you start with Example 4. Then try Examples 2 and 3 (please note the infinite outcome space - everything's nice and convergent so don't worry). And then the birthday problem in Example 5. It is useful to note that independence can be slippery when you have more than two events: see Example 8, and also look carefully at the box on page 67 – it is not enough just to check that P(ABC) = P(A)P(B)P(C).

The Chapter Summary is terrific - read it! Pages 72-73. You can ignore the bit on Odds.

CHAPTER 2

2.1: Go through pages 79-83 very thoroughly. Then read all the contents of the box on Page 86; follow that by learning the derivation using consecutive odds. Look carefully at the arrays of figures on Pages 87-89. Go through the captions and make sure you're following the details of what changes as n gets large.

2.2: Everything up to and including the box on Page 101. I will be using the terms "center" and "spread" for "mean" and "standard deviation" respectively. Also, more formally, "location parameter" and "scale parameter". The normal table is in Appendix 5.

2.4: Nice short section, read all of it.

2.5: Start on Page 124 at Sampling Without Replacement and read to the end. Again a nice short section, but be careful when you do the problems - counting can be quite slippery.

Chapter Summary: Read the portion entitled Binomial Probability Formula on Page 130. Then go over all of Page 131.

CHAPTER 3

The most important chapter in the text and in the course.

3.1: Skim pages 140-149, then read carefully anything that looks unfamiliar. In lecture we summarized most of this material. Then jump to the boxes on page 151. They will come as no surprise. Next read the first para under the heading Several Random Variables (page 153), skim the definition and consequences of mutual independence of random variables on page 154.

3.2: This section is the key to much of the rest of the course. You must go through all of it, except the Gambling Interpretation on pages 165-166 and Expectation and Prediction on pages 178-179. The summary box on page 181 is crucial. You should add to it the statement of Markov's inequality, page 174.

3.3: This is about the fundamental measure of dispersion, and like 3.2 it must be internalized deeply. Read everything except the Skewness section on page 198. Note that in lecture we will skip to Section 3.6 after this, and then come back and cover 3.4 and 3.5.

3.4: This formalizes some moves we've been making for a while, e.g. with the Poisson distribution. But the examples in this section are all in the context of the geometric distribution which is the simplest of all the distributions on an infinite set. Skim the whole section.

3.5: The Poisson is familiar as an approximation to the binomial. Here it appears in its own right as a distribution. Read pages 222-224, then 226-227. For the random scatter, read the assumptions in the boxes on page 229 and the statement of the theorem in the box on page 230. Then read the Examples 2 and 3, and the note on Thinning. You don't have to read the proof of the Poisson Scatter Theorem on page 233.

3.6: This formalizes the symmetries that you saw in card shuffling earlier in the class. Go straight to Examples 1 and 2 - you will find that you could have done them back in Chapter 2 after we talked about sampling without replacement. The main calculation is that of the mean and variance of the hypergeometric, pages 241-243.

Chapter Summary: You should be able to recite the entire summary in the way you can (I hope) recite your multiplication tables. Don't forget the box on page 181. And take a look at the final line, which points you to Distribution Summaries in the back of the text. These summaries are in my view the best available in any probability text. Students refer to them long after they have completed Stat 134.

CHAPTER 4

4.1. Pages 260-271. Follow the examples closely.

4.2. Pages 278-282, continue reading the numerical illustration at the top of page 283. Then page 284-289; understanding the summary on page 289 is important. Example 4 on page 290 is instructive because it shows how you can use gamma facts known from the Poisson process context in a setting where there appears to be no Poisson process.

4.4. Read the whole section (it's short) and follow the examples carefully.

4.5. We defined the c.d.f. very early, when we started 4.1. So skim pages 311-314, then go over Examples 1 and 2 and notice that that we did Example 1 in class when we talked about the uniform density. The discussion of max and min on pages 316-318 will come as no surprise.

4.6. Nice short section, read it all. Remember that identifying a beta distribution is easy – the density has to look like x to a power times (1-x) to another power, for x between 0 and 1. The rest is just the constant that makes the density integrate to 1.

Chapter Summary: Everything on pages 332-333 except the sections on hazard rates.

Note: Gamma densities are sprinkled throughout the chapter, in particular in Sections 4.2 and 4.4; the gamma constant also appears in Section 4.6.

CHAPTER 5

5.1. Easy but important; go through all the examples. This section gives you practice in representing events as regions in the plane.

5.2. This one has all the fundamentals, so read everything. You will find that the examples are related to the ones in lecture, so read the statement of a problem in an example and try to do it yourself before reading the solution. It is important to compare the tables on pages 348-349 line by line. You will find that you already learned all the joint density facts in Chapter 3, provided you replace sums by integrals.

5.3. This is perhaps the most important distribution in the subject. Read pages 357-361 (you will recognize the results from lecture). Then read the result in the box on page 363. The result is crucial (sums of independent normals are normal) and simple to remember, even if you choose not to go through its derivation. Example 2 on page 364 involves an important technique. Before you read about the chi-squared distribution, go back to the discussion of the gamma function for half-integer values of r. Then read the chi-square section from page 364 to just below equation (2) on page 365 where the chi-squared distribution is defined. You can ignore the rest unless you have already taken a statistics class which covered chi-squared tests.

5.4. We will cover distributions of sums, pages 371-382. The ratio example is great but everything that I ask you to do with ratios can be done without the density of the ratio, so I have omitted the density calculation.

Chapter Summary: Nice and short, go over all of it. At this point you should go through the Distribution Summaries (pages 476-478) and notice that you know all the distributions, apart from the bivariate normal which you will meet in Chapter 6. These summaries are a wonderful part of this text; you won't find this information so succintly displayed elsewhere.

CHAPTER 6

6.1. This is essentially just one example, to get you back into thinking about discrete joint distributions. Notice that, as with many examples in conditioning, it's easy to find conditional distributions if you go in "chronological order". For example, it's usually easy to find the conditional distribution of the number of heads (Y) given the number of coins (X). It takes more work to "go backwards in time", that is, to find the distribution of the number of coins given the number of heads. You need to renormalize by the probability of what's given; that is, you have to use the division rule. That's what this section is about.

6.2. Conditional expectation is a powerful tool for finding expectations. The key is the box on Page 403. Skim pages 40-403, then read Examples 2 and 3. Then go to Page 406, which formalizes a natural idea.

6.3. Start with the box on page 417, in the context of coin-tossing. Read Example 3 and Problem 1 of Example 4. Now go over the boxes on pages 410 and 411, and then look at the calculations at the bottom of page 411 to reassure yourself that a conditional density is just an ordinary density, and can be used like any other density. The diagrams on page 412, and their companion text on page 413, are terrific for a geometric understanding of the division involved in the formula for the conditional density. Go on to Example 1 and follow it thoroughly. Then look at the box on page 416 and go over Example 2. Finally, compare pages 424 and 425. They should show you that everything you know about continuous conditioning is an extension of what you already knew about discrete conditioning.

6.4. We will first use covariance as a tool to find the variance of a sum. So start with pages 430-431, then jump to page 441-444. Next comes correlation, which is the measure that gives some meaning to covariance. Go over the boxes on pages 432-433. Example 4 of the text is similar to one of the exercises on that page. Example 6 brings together all the techniques you have recently learned. It's well worth going through.

6.5. The bivariate (and multivariate) normal is the fundamental distribution of statistics. Read page 449, then the box on page 451. If you don't like the geometry of the construction of the bivariate normal, never mind (though pages 452-453 are among the best descriptions of the geometry at this level). But you must follow everything on pages 454-461. Much of it will be done in class exactly as in the text, but you must fill in the blanks.

Chapter Summary: This lists all the general formulas, but in my experience students understand these formulas much better in the context of specific examples. If a formula seems mysterious, an excellent exercise is to go through your notes and the text to find one specific example of the use of that formula.