2. Roll a pair of unbiased dice. Let X be the maximum of the two faces...

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 ?

Solutions

Expert Solution

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

orchestra answered 1 year ago

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