Page 190 - Probability and Statistical Inference

3. Multivariate Random Variables 167

3.5.6 (Example 3.5.4 Continued) Construct some other specific functions
f (x ), i = 1, 2, 3 satisfying the conditions laid out in the Example 3.5.4 and
i
i
observe what happens in some of those special situations.
3.5.7 Let us start with the non-negative integrable functions f (x ) for 0 < x i
i
i
< 1 which are not identically equal to unity such that we can claim:
f (x )dx = 1, i = 1, 2, 3, 4. With x = (x , x , x , x ), let us then define
i i i 1 2 3 4

for 0 < x < 1, i = 1, 2, 3, 4.
i
(i) Directly by integration, check that g(x , x , x , x ) is a pdf;
1 2 3 4
(ii) Directly by integration, find the expression of the marginal pdf
g (x , x , x );
1, 2,3 1 2 3
(iii) Show directly that X , X , X are independent;
1 2 3
(iv) Consider the random variables X , X , X , X whose joint pdf is
4
1
3
2
given by g(x , x , x , x ). Show that X , X , X , X are not indepen
2
4
3
2
3
1
1
4
dent.
3.5.8 (Exercise 3.5.7 Continued) Along the lines of the Exercise 3.5.6,
construct some specific functions f (x ), i = 1, 2, 3, 4 satisfying the conditions
i
i
laid out in the Exercise 3.5.7 and observe what happens in some of those
special situations.
3.5.9 Prove the Theorem 3.5.3.
3.5.10 Suppose that X and X have the joint pmf
1 2
Prove whether or not X and X are independent.
1 2
3.5.11 Suppose that X and X have the joint pmf
1 2
Prove that X and X are dependent.
1 2
3.5.12 Suppose that X and X have the joint pmf
1 2