Page 192 - Probability and Statistical Inference

3. Multivariate Random Variables 169

(i) Show that f (x ) = ¼π{cos(½πx ) + sin(½πx )} for 0 < x < 1,
1 1 1 1 1
and f (x ) = ¼π{cos(½πx ) + sin(½πx )} for 0 < x < 1;
2 2 2 2 2
(ii) Prove whether or not X and X are independent.
1 2

Figure 3.10.4. Plot of the PDF from the Exercise 3.5.16
3.5.17 Suppose that the random vector X = (X , X , X , X ) has its joint
1
2
3
4
pdf given by

(i) Find the marginal pdf’s f (x ), i = 1, 2, 3, 4. Show that each X is
i
i
i
distributed as N(0, 4), i = 1, 2, 3, but X is distributed as N(0, 1);
4
(ii) By integrating f(x , x , x , x ) with respect to x alone, obtain the
1
2
4
3
4
joint pdf of (X , X , X );
1 2 3
(iii) Combine the parts (i) and (ii) to verify that X , X , X form a set of
1
3
2
independent random variables;
(iv) Show that X , X , X , X do not form a set of independent random
1 2 3 4
variables.
3.5.18 (Exercise 3.5.17 Continued) Consider the random vector X = (X ,
1
X , X , X ) with its joint pdf given in the Exercise 3.5.17. Using this random
4
3
2
vector X, find few other four-dimensional random vectors Y = (Y , Y , Y , Y )
3
4
2
1
where Y , Y , Y are independent, but Y , Y , Y , Y are not.
1 2 3 1 2 3 4
3.5.19 Suppose that X has the following pdf with –∞ < µ < ∞, 0 < σ < ∞: