Question

In: Statistics and Probability

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

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