Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 198

Answer From ( 3.238 ), we have:

PfX 1 ¼ x 11 g¼ X

3

PfX 1 ¼ x 11 ; X 2 ¼ x 2 j g¼ 0 : 1 þ 0 : 15 þ 0 : 2 ¼ 0 : 45 :

j¼ 1

PfX 1 ¼ x 12 g¼ X

3

PfX 1 ¼ x 12 ; X 2 ¼ x 2 j g¼ 0 : 15 þ 0 : 25 þ 0 : 15 ¼ 0 : 55 : (3.239)

j¼ 1

The marginal distribution and density are, respectively:

F X 1 ðx 1 Þ¼ 0 : 45 uðx 1 1 Þþ 0 : 55 uðx 1 þ 1 Þ; (3.240)

f X 1 ðx 1 Þ¼ 0 : 45 dðx 1 1 Þþ 0 : 55 dðx 1 þ 1 Þ: (3.241)

Similarly,

PfX 2 ¼ x 21 g¼ X

2

PfX 1 ¼ x 1 j ; X 2 ¼ x 21 g¼ 0 : 1 þ 0 : 15 ¼ 0 : 25 ;

j¼ 1

PfX 2 ¼ x 22 g¼ X

2

PfX 1 ¼ x 1 j ; X 2 ¼ x 22 g¼ 0 : 15 þ 0 : 25 ¼ 0 : 4 ;

j¼ 1

PfX 2 ¼ x 23 g¼ X

2

PfX 1 ¼ x 1 j ; X 2 ¼ x 23 g¼ 0 : 2 þ 0 : 15 ¼ 0 : 4 ; (3.242)

j¼ 1

F X 2 ðx 2 Þ¼ 0 : 25 uðx 2 Þþ 0 : 4 uðx 2 2 Þþ 0 : 35 uðx 2 5 Þ: (3.243)

f X 2 ðx 2 Þ¼ 0 : 25 dðx 2 Þþ 0 : 4 dðx 2 2 Þþ 0 : 35 dðx 2 5 Þ: (3.244)

Exercise 3.6 Find the conditional distributions,

F X 1 x 1 jX 2 ¼ x 21 ¼ 0 ð Þ; and F X 2 x 2 jX 1 ¼ x 11 ¼ 1 ð Þ; (3.245)

for the random variables from Exercise 3.5.

Answer From ( 3.46 ), we have:

P

2
i ¼ 1 PfX 1 ¼ x 1 i ; X 2 ¼ x 21 ¼ 0 guðx 1 x 1 i Þ
PfX 2 ¼ x 21 ¼ 0 g

F X 1 x 1 jX 2 ¼ x 21 ¼ 0 ð Þ ¼ : (3.246)

From ( 3.238 ), we have:

PX 1 ¼ 1 f ; X 2 ¼ 0 g ¼ 0 : 1 ; PX 1 ¼ 1 f ; X 2 ¼ 0 g ¼ 0 : 15 : (3.247)