Page 185 - Probability and Statistical Inference

162 3. Multivariate Random Variables

Figure 3.10.1. Plot of the PDF from the Exercise 3.2.14

3.3.1 (Examples 3.3.1-3.3.2 Continued) Evaluate E(X ) and V(X ). Also,
evaluate µ and 1 1
1/2
3.3.2 (Examples 3.3.3 Continued) Answer the following questions.
(i) Evaluate E(X ) and V(X ), i = 1, 2 using (3.3.19);
i i
(ii) Evaluate E[X (1 – X )], and E[(X + X ) ];
2
1 1 1 2
(iii) Show that f (x ) = f (x ) for i ≠ j = 1, 2.
i|j i i i
3.3.3 (Example 3.3.5 Continued) Evaluate E(X ) and V(X ), i = 1, 2. Evalu-
ate µ and for i ≠ j, i, j = 1, 2. Also evaluate i i
i|j

3.3.4 Let c be a positive constant such that X and X have the joint pdf
given by 1 2

Find the value of c. Derive the expressions of f (x ), f (x ), f (x ), and
1
1
2
1/2
1
2
f (x ). Evalute E(X ) and V(X ), i = 1, 2. Evaluate µ and for i ≠ j,
i|j
2
2/1
i
i
i, j = 1, 2. Also evaluate E[X (1 – X )], and E[(X +
1 1 1