Class summary:
Monday, 3/17
I introduced another important method for computing expectations,
the so-called indicator method, and worked several examples
illustrating the use of this method:
(i) the expectation of the binomial(n,p)
distribution; (ii) the expected number of red balls in a sample of
n balls taken out of R red and B black balls, without replacement
(this is the standard box/ball problem that came up in connection with
combinatorial probabilities); and (iii) the expected number of distinct
birthdays in the birthday problem. The key to applying this method is to
identify events Ai such that the r.v. X is equal to the number of Ai's
occurring.
Reading Guide to the Pitman Text: Sections 3.1 and 3.2
3.1 Random variables: Introduction
A random variable is an important concept in probability theory.
This section provides an excellent introduction
to, and motivation of, this abstract concept, and you should carefully
study this section (except for those parts that have been explicitly
skipped). The examples in this section complement well the examples I
worked in class, and they serve as models for
some of the homework problems.
- p.139 - 141 (Introduction and Distribution of X): Read this
part. It's an easy read, and has some simple but instructive examples.
- p.141, bottom - 143: You can skip most of
this part, to the extent that it involves a function g(W). However, the
examples on p. 143 are simple illustrations of "Type II" problems (as
defined in the lecture of 3/22).
- p.144 - 150, top (joint distributions): Study this part, and in
particular Examples 2 - 4. The boxed formulas on top of p. 147 are
illustrations of type II probability computations.
- p.150, middle - end of section (Conditional distributions,
Several random variables, Symmetry): Skip
these sections except for the definitions
of independence of random variables
(see the bottom of p. 151 for the case of two r.v.'s and middle of p.154
for more than two r.v.'s)
- Exercises 3.1
- 1. Quickie
- 2. An instructive problem that was (essentially) done in
class.
- 3. An easy problem, but (b) is somewhat tedious (unless
you spot the pattern).
- 4. Part of HW 6.
- 6. Part of HW 6.
- 7. This boils down to a familiar exercise in set-theoretic
probability.
- 9. Part of HW 6.
- 13(a). Part of HW 6.
- 14. Part of HW 6.
An interesting and instructive problem, and one of the
problems included in the Problem Sampler
handed out at the beginning of class.
- 22. Somewhat unconventional, but easier than it looks.
Just use the definitions of independence.
3.2 Expectation
The expectation E(X) is a numerical quantity associated with a r.v. X,
defined by the formula on p.163, which
represents an average of all values of the r.v., weighted
by the probabilities with which these values are taken on.
The expectation of a r.v. behaves much like an integral of a function.
An important interpretation/application of the expectation is as the
"fair value" or "break-even price" of a game of chance.
The expectation can be computed (i) via the definition, (ii)
indirectly using rules of expectations, and (iii) by the indicator
method.
Which method one should use is usually dictated by the context.
The presentation of expectation in the text is excellent, and you should
read all of Section 3.2, except for the following parts:
- p. 171, bottom - 174: Skip the tail sum formula on p. 171,
Boole's inequality on p. 173, and Markov's inequality on p. 174.
However, the first part of Example 9, and the "Discussion" following
that example, is instructive as it illustrates the minimum trick.
- p. 178 - 179: Skip the part on "Expectation and Prediction."
- Exercises 3.2
- 1. Quickie
- 3. Quickie
- 5. A practical application of the "fair price"
interpretation of an expectation. The calculations are not difficult,
but somewhat tedious, as they break down into many cases.
- 7. Use the indicator method.
- 8. Part of HW 7.
- 9. Routine. Just use the properties of the expectation.
- 10. This is similar to, though somewhat simpler than,
3.3.8, which is part of HW 7. Not particularly hard - just use the
definition of an indicator random variable. The key is to understand
what the product of two indicator r.v.'s, IA IB,
is.
- 13. Part of HW 7.
- 14. Another case for the indicator method. Of the same type
as the birthday problem done in class.
Be sure to study the formula summary on p. 181 and memorize those
formulas; this summary complements the end-of-chapter summary on p. 248
- 249.
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