Math 361 class summary, Friday, 2/14

Class summary: Friday, 2/14

I introduced two important approximations to the binomial distribution, the Poisson approximation and the normal approximation. The Poisson approximation is best suited in the case of rare events and a large number of trials. while the normal approximation works well when the number of trials is More precisely, in order for the Poisson approximation to be accurate, p should be very small, n should be large (ideally of order 1/n), and k should be small (which it almost always is, since the cases of interest are those when k is 0, 1, 2, or some other very small number). If these conditions are not satisfied, the Poisson approximation should not be applied, and it may be completely off. By contrast, for the normal approximation to work well, p should not be too small or too close to 1, and k should be near the expected value (namely, pn).

I distributed three handouts: One on Birthday Type Problems, containing two problems motivated by the original birthday problem, which can be done within the success/failure framework; a handout on the normal approximation, and a handout with a comparison of the normal and the Poisson approximation, which illustrates how the normal and Poisson approximations compare in some particular cases. The normal approximation handout contains a simple set of formulas, that I would recommend using instead of those presented in the text. The formulas given on this handout are all you need to know to solve problems involving normal distribution. Stick to these formulas (there are three altogether for normal approximation, along with defining formulas for phi,Phi, mu, and sigma), rather than trying to apply one of the many formulas given in Section 2.2 of the book.

Reading Guide

Section 2.2: The normal approximation

This is one of the most important topics in this course, and you should make sure that you fully understand this material. For the relevant formulas, refer to the Normal Distribution Handout instead of the various formulas given in this section. Some of the formulas given in the book involve a "1/2 correction" which can make the approximation slightly more accurate in some cases, at the expense of complicating the calculations, but which in most cases of interest does not make much of a difference. The formulas on the handout do not have the correction and they are all you need to know to do normal approximation problems.

Graphs: The graphs on p. 96 - 98 illustrate nicely the normal distribution and how this distribution approximates the binomial distribution.

Example 1: This example was (with different numbers) done in class using the first formula on the normal approximation handout; the method given in the text is much more complicated, so stick to the method presented in class.

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. The square root "law" will not be on the exam.

Confidence intervals, p. 101-102. You should not memorize specific probabilities for confidence intervals, but you should know how to derive confidence intervals from general formulas for normal approximation. (The derivation is quite easy and was done in class on 2/19.)

Examples 2 - 4: These are instructive examples in applying confidence intervals. Examples 2 and 4 deal with the situation in which p is unknown, as in the case of opinion polls. (See the class of 2/21 for setting up a S/F model for opinion polls.)

Skew-normal approximation, p. 103 - end of section: You can skip this part.

Exercises 2.2:

1,3: This is a quickie.
6: Part of HW 4. A routine application of normal approximation.
8:A quickie.
9 (airline overbooking): This was essentially done in class.
10: Part of HW 4.
11 (batting averages): An instructive, but not too difficult problem.
12: An instructive but not very hard problem, related to the derivation of confidence intervals.
13: A problem on opinion polls and confidence intervals.
14: Part of HW 4. Part (a) is completely routine; part (b) is (somewhat) similar to the airline overbooking problem.

Section 2.3: Derivation of the normal approximation

Read or skim through this section if you are interested in why normal approximation works. This section will not be on the exam or required for HW problems.

Section 2.4: The Poisson approximation

The Poisson distribution, given on p. 121, is one of many "named" distributions (the binomial distribution and the geometric distribution are other examples). Its significance lies in the fact that it can be used to approximate the binomial distribution; the formula for this approximation is given on p. 119. It is important to keep in mind that this approximation is only good if certain conditions are met; namely, p should be small, n large (typically, of order 1/p), and k (which denotes the number of successes in the binomial distribution) small (typical values for k being 0, 1, or 3). If these conditions are not satisfied, the Poisson approximation is not reliable (it may be off by several orders of magnitude) and should not be used.

p. 117: The three graphs of binomial distributions with different values of n and p (related by p=1/n in all three cases) illustrate the Poisson approximation. The graphs are nearly identical, and if the graph of the Poisson distribution with parameter mu = 1 had been plotted as well, it would be also nearly indistinguishable from these three graphs. This confirms the fact that (under appropriate conditions, which are clearly satisfied here), a binomial distribution is well approximated by the Poisson distribution with parameter mu = np (which is 1 in this case).
p.118: Here some theoretical background behind the Poisson approximation is presented. You are not expected to know this material and you can therefore skip that page, but you might want to skim through the page if you are curious as to why the Poisson approximation works.
Example 4: This is a simple, but instructive example which you should study. We did a similar example in class.
Formulas: You need to memorize the boxed formulas for the Poisson distribution on p. 201, and for the Poisson approximation on p.119. (The latter is simply the formula for the Poisson distribution with a particular choice of mu, namely mu = n p).
Graphs: The bar graphs ("histograms") of the binomial and Poisson distribution on p. 117 and p. 120 give you an idea of what these distributions look like with various choices of the parameters involved (n, p, mu).
Exercises 2.4 Problems 2 - 5 are easy quickies, similar to Example 4.

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