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3. Multivariate Random Variables 163

X ) ]. The surface represented by this pdf is given in the Figure 3.10.2.
2
2

Figure 3.10.2. Plot of the PDF from the Exercise 3.3.4 with c = 3/16

3.3.5 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 f (x ).
2
1
1
1
2/1
2
2
1/2
Does either of X , X have a standard distribution which matches with one of
1
2
those distributions listed in Section 1.7? Evaluate E(X ) and V(X ), i = 1, 2.
i
i
Evaluate µ and for i ≠ j, i, j = 1, 2. Also evaluate E[X (1 – X )],
i/j 1 1
2
and E[(X + X ) ]. The surface represented by this pdf is
1
2
given in the Figure 3.10.3.

Figure 3.10.3. Plot of the PDF from the Exercise 3.3.5 with c = 1

3.3.6 (Example 3.3.10 Continued) Evaluate E[X X ] and E[X Also
1
2
evaluate E[X (1 – X )], E[X X ] and E[X + X + X ) ]. 1
2
1 1 1 2 1 2 3
3.3.7 (Example 3.3.10 Continued) Find the expressions for E[X 1
| X = x , X = x ] and E[X | X = x ] with 0 < x < 2, 0 < x < 1.
2 2 3 3 1 2 2 2 3