Page 191 - Probability and Statistical Inference

168 3. Multivariate Random Variables

Prove whether or not X and X are independent.
1 2
3.5.13 Suppose that X and X have the joint pmf such that P{X = 2 ∩ X 2
1
2
1
= 3} = 1/3, P{X = 2 ∩ X = –1} = a, P{X = –1 ∩ X = 3} = b and P{X =
1
2
1
2
1
–1 ∩ X = –1} = 1/6 where a and b are appropriate numbers. Determine a and
2
b when X and X are independent.
1 2
3.5.14 Suppose that X and X have the joint pdf
1 2
where k is a positive constant.
(i) Show that k = 2;
(ii) Show that f (x ) = 2{e – e –2x 1 } I(0 < x < ∞) and f (x ) = 2e –2x 2
–x
1
1
2
2
1
1
I(0 < x < ∞). Find P(X ≥ 3);
2 2
(iii) Show that f (x ) = e –(x – x ) I(x < x < ∞) and f (x ) = e –x 2
2/1
1
2
2
1
2
1/2
1
–x
–1
(1 –e ) I(0 < x < x );
1 2 1
(iv) Prove whether or not X and X are independent.
1 2
3.5.15 Suppose that X and X have the joint pdf
1 2
(i) Show that f (x ) = ¼(3 – x ) I(0 < x < 2) and f (x ) = ¼(5 – x ) I(2
1 1 1 1 2 2 2
<x < 4);
2
(ii) Show that f (x ) = I(0 < x < 2, 2 < x < 4) and
1/2 1 1 2
f (x ) = I(0 < x < 2, 2 < x < 4);
2/1 2 1 2
(iii) Prove whether or not X and X are independent.
1 2
3.5.16 Suppose that X and X have the joint pdf
1 2

The surface represented by this pdf is given in the Figure 3.10.4.