Class summary:
Friday, 3/7
I introduced the concept of independence in the context of random
variables: two r.v.'s X and Y are independent if the joint
distribution is the product of the individual distributions at all
values x of X and y of Y. This is related to (but not the same as)
independence of two events.
I then discussed, more generally, the
relation between the joint distribution of two random variables, X and Y, and the
individual ("marginal") distributions of each of these random variables.
The individual distributions can always be computed from
the joint distribution, but the converse is not true without additional
information about the interplay between X and Y. One such piece of
additional information is that X and Y are independent; in that case (and
only then), the joint distribution can be obtained as the product
of the individual distributions.
In the second part of the hour
I did some examples of problems on random variables. Such problems
come in two varieties:
(I) Given a description
of the random variable(s), find the distribution (or joint
distribution). (II) Given a
distribution of a r.v., (or a joint distribution of two r.v.'s),
compute probabilities involving these r.v.'s.
Problems of type (I) boil down to classical probability computations of the
type done in Chapters 1 and 2. To solve problems of type (II), express the
probability asked for as a sum over individual probabilities
P(X=x) (or P(X=x,Y=y)), where the summation condition mirrors the condition
on X and Y in the probability sought.
Reading guide, Section 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 3/7).
- 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.
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