Page 187 - Probability and Statistical Inference

164 3. Multivariate Random Variables

{Hints: First find the conditional pdf’s of X given X = x , X = x and that of
3
3
2
1
2
(X , X ) given X = x . Next, express the expected values as the respective
1 3 2 2
integrals and then evaluate them.}
3.3.8 (Example 3.3.10 Continued) Choose any three specific pdf’s on three
finite intervals as a (x ), i = 1, 2, 3. Then, construct an appropriate combina-
i
i
tion of these a ’s to define a pdf f(x) along the line of (3.3.47). Try putting
i
these components in a certain order and adjust the coefficients so that the
whole integral is one. {Hint: Examine closely why we had the success in the
equation (3.3.48).}
3.3.9 (Example 3.3.11 Continued) Evaluate E[X X X ] and E[ X ].
1
5
2
5
Also evaluate E[X (1 – X )], E[X X (2 – X ) ] and E[(X + X + X + X +
2
4
3
2
1
5
1
3
2
1
2
X ) ].
5
3.3.10 (Example 3.3.11 Continued) Find the expressions for E[X (1 – X )
5
1
| X = x , X = x ] and E[X | X = x ] with 0 < x < 2, 0 < x , x < 1. {Hints:
4
1
5
5
2
4
5
2
2
4
First find the conditional pdf’s of (X , X ) given X = x , X = x and that of
5
1
2
4
4
2
(X , X ) given X = x . Next, express the expected values as the respective
1
5
5
3
integrals and then evaluate them.}
3.3.11 (Example 3.3.11 Continued) Choose any five specific pdf’s on five
finite intervals as a (x ), i = 1, ..., 5. Then, construct an appropriate combina-
i
i
tion of these a ’s to define a pdf f(x) along the line of (3.3.53). Try putting
i
these components in a certain order and adjust the coefficients so that the
whole integral is one. {Hint: Examine closely how we had defined the func-
tion f in the equation (3.3.53).}
3.3.12 Let c be a positive constant such that X , X and X have the joint
2
3
1
pdf given by
Find the value of c. Derive the marginal pdf’s of X , X and X . Does either of
1
3
2
X , X , X have a standard distribution which matches with one of those
3
1
2
distributions listed in Section 1.7? Evaluate E[X X ] and E[X ]. Also
1
1
2
2
evaluate E[X (1 – X )], E[X X ] and E[(X + X + X ) ].
1 1 1 2 1 2 3
3.4.1 (Example 3.2.1 Continued) In the case of the random variables X ,
1
X whose joint pdf was defined in the Table 3.2.1, calculate ρ, the correlation
2
coefficient between X and X .
1 2
3.4.2 (Exercise 3.2.4 Continued) In the case of the random variables X ,
1
X whose joint pdf was defined in the Exercise 3.2.4, calculate ñ, the correla-
2
tion coefficient between X and X .
1 2