Page 188 - Probability and Statistical Inference

3. Multivariate Random Variables 165

3.4.3 (Example 3.3.10 Continued) Evaluate Cov(X , X ), i ≠ j = 1, 2, 3.
j
i
Then, evaluate ρ , i ≠ j = 1, 2, 3.
Xi, Xj
3.4.4 (Example 3.3.11 Continued) Evaluate Cov(X , X ), i ≠ j = 1, ..., 5.
i
j
Then, evaluate ρ , , i ≠ j = 1, ..., 5.
Xi Xj
3.4.5 Prove the Theorem 3.4.3.
3.4.6 With any two random variables X and X , show that
1 2
provided that Cov(X , X ), V(X ), and V(X ) are finite.
1 2 1 2
3.4.7 (Exercise 3.4.6 Continued) Consider any two random variables X
and X for which one has V(X ) = V(X ). Then, for this pair of random vari- 1
2
2
1
ables X and X we must have Cov(X , X ) = 0. Next, using this ammunition,
1
2
2
1
construct several pairs of random variables which are uncorrelated.
3.4.8 (Examples 3.2.6-3.2.7 Continued) Consider the multinomial random
variable X = (X , ..., X ) defined in the Example 3.2.6. Evaluate ρ , for all i
6
1
Xi Xj
≠ j = 1, ..., 6.
3.4.9 (Example 3.2.8 Continued) Consider the multinomial random vari-
able X = (X , X , ) defined in the Example 3.2.6 with = n – (X + X ).
1
1
5
5
Evaluate ρ , , ρ , and ρ , .
X1 X5 X1 X5
3.4.10 (Exercise 3.2.7 Continued) Consider the multinomial random vari-
able X = (X , X , X , X ) defined in the Exercise 3.2.7. Evaluate ρ , for all i
3
Xi Xj
4
1
2
≠ j = 1, 2, 3, 4.
3.4.11 (Exercise 3.2.10 Continued) Recall that X = (X , ..., X ) has the
5
Mult (n, p , ..., p ) with 0 < p < 1, i = 1, ..., 5, 1 whereas Y = (Y ,
1
i
5
1
5
Y , Y ) where Y = X + X , Y = X , Y = X + X . Evaluate ρ , for all i ≠ j =
1
4
2
3
5
Yi Yj
2
2
1
3
3
1, 2, 3.
3.4.12 Suppose that X and X have the joint pdf given by
1 2
(i) Show that f (x ) = x + 1/2, f (x ) = x + 1/2 for
1 1 1 2 2 2
0 < x , x < 1;
1 2
(ii) Evaluate P{1/2 ≤ X ≤ 3/4 | 1/3 ≤ X ≤ 2/3}.
1 2
3.4.13 Suppose that X and X have the joint pdf given by
1 2