In: Statistics and Probability
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?
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