Page 189 - Probability and Statistical Inference

166 3. Multivariate Random Variables

Show that ρ , = – 5/139.
X1 X2

3.5.1 (Exercise 3.3.4 Continued) Let c be a positive constant such that X 1
and X have the joint pdf given by
2

Prove whether or not X and X are independent. Solve this exercise first by
1
2
directly applying the Definition 3.5.1. Then, repeat this exercise by applying
the Theorem 3.5.3.
3.5.2 (Exercise 3.3.5 Continued) Let c be a positive constant such that X 1
and X have the joint pdf given by
2

Prove whether or not X and X are independent. Solve this exercise first by
1
2
directly applying the Definition 3.5.1. Then, repeat this exercise by applying
the Theorem 3.5.3.
3.5.3 (Exercise 3.3.12 Continued) Let c be a positive constant such that
X , X and X have the joint pdf given by
1 2 3

Prove whether or not X , X and X are independent. Solve this exercise first
1
2
3
by directly applying the Definition 3.5.1. Then, repeat this exercise by apply-
ing the Theorem 3.5.3.
3.5.4 (Example 3.5.4 Continued) Verify all the steps in the Example 3.5.4.
3.5.5 (Example 3.5.4 Continued) In the Example 3.5.4, suppose that we
fix f (x ) = 2x , and , 0 < x , x , x < 1.
1 1 1 1 2 3
Then, form the function g(x , x , x ) as in (3.5.2).
1 2 3
(i) Directly by integration, check, that g(x , x , x ) is a pdf;
1 2 3
(ii) Directly by integration, find the expressions of all pairwise marginal
pdf’s g (x , x ) and single marginal pdf’s g (x ) for i ≠ j = 1, 2, 3;
i,j i j i i
(iii) Show directly that the X’s are pairwise independent, but X , X , X
1 2 3
are not independent.