In: Statistics and Probability
2. Roll a pair of unbiased dice. Let X be the maximum of the two faces and Y be the sum of the two faces. What is the joint density of X and Y ?
To find the joint distribution:
Y = 2, X = 1 (1,1)
Y = 3, X = 2 (1,2) & (2,1)
Y = 4, X = 2 (2,2) Y = 4, X = 3 (3,1) & (1,3)
Y = 5, X = 4 (4,1) & (1,4) Y = 5, X = 3 (3,2) & (2,3)
Y = 6, X = 4 (4,2) & (2,4) Y = 6, X = 3 (3,3) Y = 6, X = 5 (5,1) & (1,5)
Y = 7, X = 4 (4,3) & (3,4) Y = 7, X = 5 (5,2) & (2,5 ) Y = 7, X = 6 (6,1) & (1,6)
Y = 8, X = 4 (4,4) Y = 8, X = 5 (5,3) & (3,5 ) Y = 8, X = 6 (6,2) & (2,6)
Y = 9, X = 5 (5,4) & (4,5 ) Y = 9, X = 6 (6,3) & (3,6)
Y = 10, X = 5 (5,5) Y = 10, X = 6 (6,4) & (4,6)
Y = 11, X = 6 (6,5) & (5,6)
Y = 12, X = 6 (6,6)
X/Y | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Total |
1 | 1/36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1/36 |
2 | 0 | 2/36 | 1/36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3/36 |
3 | 0 | 0 | 2/36 | 2/36 | 1/36 | 0 | 0 | 0 | 0 | 0 | 0 | 5/36 |
4 | 0 | 0 | 0 | 2/36 | 2/36 | 2/36 | 1/36 | 0 | 0 | 0 | 0 | 7/36 |
5 | 0 | 0 | 0 | 0 | 2/36 | 2/36 | 2/36 | 2/36 | 1/36 | 0 | 0 | 9/36 |
6 | 0 | 0 | 0 | 0 | 0 | 2/36 | 2/36 | 2/36 | 2/36 | 2/36 | 1/36 | 11/36 |
Total | 1/36 | 2/36 | 3/36 | 4/36 | 5/36 | 6/36 | 5/36 | 4/36 | 3/36 | 2/36 | 1/36 | 1 |