65 Using the clues given below fill in the rest of the joint

65. Using the clues given below, fill in the rest of the joint distribution. There is only one answer: Fork = 1,2,3, P(Y= 1|X= k) = 2/3, P(X= k|Y= 1) = k/6.

Solution

The table can be represented as:

Where A, B , C, D, E and F are the missing probabilities.

Total probability = A + B + C + D + E + F = 1

P(Y=1 | X=1) = A / (A +D) = 2/3

So A = 2D

P(Y=1 | X=2) = B / (B +F) = 2/3

So B = 2F

P(Y=1 | X=3) = C / (C +E) = 2/3

So C = 2E

P(X=1 | Y=1) = 1/6

So A / (A + B +C) = 1/6

P(X=2 | Y=1) = 2/6

So B / (A + B +C) = 2/6

P(X=3 | Y=1) = 3/6

So C / (A + B +C) = 3/6

From above 3 equations: B = 2A and C = 3A

D = A/2 ; F = B/2 = A ; E = C/2 = 3A/2

A + 2A + 3A + A/2 + A + 3A/2 = 1

A = 1/9

B = 2/9

C = 3/9 = 1/3

D = 1/18

E = 1/6

F = 1/9

Answer:

Y	X=1	2	3
1	A	B	C
2	D	0	E
3	0	F	0