Question

If the joint probability distribution of X and Y is given by:

f (x, y) = 3k (x + y), for x = 0, 1, 2, 3; y = 0, 1, 2.

a) .- Find the constant k.

b) .- Using the table of the joint distribution and the marginal distributions, determine if variable X and variable Y are independent.

Answer 1

(a)

For a valid pdf, sum of probabilities must be equal to 1. Following table shows the calculations:

X	Y	3(x+y)	3k(x+y)
0	0	0	0
0	1	3	3k
0	2	6	6k
1	0	3	3k
1	1	6	6k
1	2	9	9k
2	0	6	6k
2	1	9	9k
2	2	12	12k
3	0	9	9k
3	1	12	12k
3	2	15	15k
Total		90	90k

Since sum of probabilities must be equal to 1 so

90k = 1

k = 1 /90

That is pdf will be

Following table shows the joint and marginal pdfs:

			X
		0	1	2	3	P(Y=y)
	0	0	1/30	2/30	3/30	6/30
Y	1	1/30	2/30	3/30	4/30	10/30
	2	2/30	3/30	4/30	5/30	14/30
	P(X=x)	3/30	6/30	9/30	12/30	1

b)

If X and Y are independent then following must be true for each X and Y

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

From table we have

P(X=0, Y=0) = 0

P(X=0) = 3/30, P(Y=0) = 6/30

Since P(X=0, Y=0) is not equal to P(X=0)*P(Y=0) so X and Y are not independent.