Question

Suppose you roll two 6 sided dice, letting X be the sum of the numbers shown on the dice and Y be the number of dice that show an odd number.

a) Find the joint pmf of <X,Y>

b) Find the marginals pmf's for both variables.

c) Are X and Y independent?

Answer 1

X takes values 2-(1,1),

3-(1,2),(2,1)

4-- (1,3),(3,1),(2,2)

5--(1,4),(4,1),(2,3),(3,2),

6--(1,5),(5,1),(2,4),(4,2),(3,3)

7--(1,6),(6,1),(2,5),(5,2),(3,4),(4,3)

8--(2,6),(6,2), (3,5),(5,3),(4,4)

9--(3,6),(6,3),(4,5),(5,4)

10--(4,6),(6,4),(5,5)

11--(5,6),(6,5)

12--(6,6)

Y takes values 0--(2,2),(2,4)(2,6),(4,2),(4,4),(4,6),(6,2),(6,4),(6,6)

1--(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),(3,6),(4,1),(4,3),(4,5),(5,2),(5,4),(5,6),(6,1),(6,3),(6,5)

2--(1,1),(1,3),(1,4),(3,1),(3,3),(3,5),(5,1),(5,3),(5,5)

So joint pmf of X ,Y

						X
		2	3	4	5	6	7	8	9	10	11	12
	0			0.03		0.055556		0.083333		0.055556		0.027778	0.25
Y	1		0.055556		0.111111		0.166667		0.111111		0.055556		0.5
	2	0.027778		0.055556		0.083333		0.055556		0.027778			0.25
		0.027778	0.055556	0.083333	0.111111	0.138889	0.166667	0.138889	0.111111	0.083333	0.055556	0.027778	1

Marginal pmf of X

X : 2 3 4 5 6 7 8 9 10 11 12

P(X):0.027778 0.055556 0.083333 0.111111 0.1388889 0.166667 0.138889 0.11111  0.0833333 0.055556 0.027778

Marginal pmf of Y

Y: 0 1   2

P(Y):0.25 0.5 0.25

P(X=4,Y=0) = 0.03

P(X=4) = 0.083333

P(Y=0) = 0.25

P(X=4).P(Y=0) = 0.02

P(X=4,Y=0) not equal to P(X=4).P(Y=0)

X and Y are not independent