Page 194 - Probability and Statistical Inference

3. Multivariate Random Variables 171

for –∞ < x , x , x < ∞. Show that X , X , X are dependent random variables.
2
3
3
2
1
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Also show that each pair (X , X ), (X , X ) and (X , X ) is distributed as a
1 2 2 3 1 3
bivariate normal random vector.
3.7.1 This exercise provides a discrete version of the Example 3.7.1. Sup-
pose that a random variable X has the following probability distribution.
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X values: –5 –2 0 2 5
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Probabilities: .25 .2 .1 .2 .25
Define Then, show that
(i) E[X ] = ] = 0;
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(ii) Cov(X , X ) = 0, that is the random variables X and X are
1 2 1 2
uncorrelated;
(iii) X and X are dependent random variables.
1 2
3.7.2 (Exercise 3.7.1 Continued) From the Exercise 3.7.1, it becomes
clear that a statement such as “the zero correlation need not imply indepen-
dence” holds not merely for the specific situations handled in the Examples
3.7.1-3.7.2. Find a few other examples of the type considered in the Exercise
3.7.1.
3.7.3 In view of the Exercises 3.7.1-3.7.2, consider the following gener-
alization. Suppose that X has an arbitrary discrete distribution, symmetric
1
about zero and having its third moment finite. Define . Then, show that
(i) E[X ] = E[ ] = 0;
1
(ii) Cov(X , X ) = 0, that is the random variables X and X are
1 2 1 2
uncorrelated;
(iii) X and X are dependent random variables.
1 2
3.7.4 (Example 3.7.1 Continued) In the Example 3.7.1, we used nice tricks
with the standard normal random variable. Is it possible to think of another
example by manipulating some other continuous random variable to begin
with? What if one starts with X that has a continuous symmetric distribution
1
about zero? For example, suppose that X is distributed uniformly on the
1
interval (–1, 1) and let Are X and X uncorrelated, but dependent?
1
2
Modify this situation to find other examples, perhaps with yet some other
continuous random variable X defined on the whole real line ℜ and
1
3.7.5 In the types of examples found in the Exercises 3.7.1-3.7.4, what if
someone had defined instead? Would X and X thus
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defined, for example, be uncorrelated, but dependent in the earlier exercises?