Question

The joint distribution of X and Y is given by |x/y 1 2 3 1 0.06 0.42 0.12 2 0.04 0.28 0.08 1. Are X and Y independent or dependent? 2. Prove your answer in part 1.

Answer 1

Solution:

The joint distribution of X and Y is

X/Y	1	2	3
1	0.06	0.42	0.12
2	0.04	0.28	0.08

The marginal distribution of X is

X 1. 2 Total

P(X=x) 0.60. 0.40. 1

The marginal distribution of Y is

Y. 1. 2. 3. Total

P(Y= y). 0.10. 0.70. 0.20. 1

If two variables X and Y are said to be independent random variables if

P( X= x, Y= y) = P( X=x) . P( Y = y)

that is $\ Pij = Pi.* P.j \ \forall (i,j)$

Case 1) i= 1 and j= 1

P11 = 0.06 and P1. = 0.60

P.1 = 0.10

P1.* P.1 = 0.60*0.10

P1.*P.1= 0.06

P11= P1.*P.1

Case2 ) i= 1 and j = 2

P12= P1.*P.2

0.42 = 0.60* 0.70

0.42 = 0.42

P12 = P1.*P.2

Case3) i= 1 and j = 3

P13 = P1.* P.3

0.12 = 0.60* 0.20

0.12 = 0.12

P13 = P1.*P.3

Case 4) i= 2 and j = 2

P22= P2.* P.2

0.28 = 0.40* 0.70

0.28= 0.28

P22 = P2.*P.2

Case5) i = 2 and j = 3

P23 = P2.*P.3

0.08 = 0.40* 0.20

0.08 = 0.08

P23 = P2.*P.3

$Here \ Pij = Pi.* P.j \ \forall (i,j)$

Hence X and Y are independent random variables