For a valid probability distribution sum of probabilities must be equal to 1. Following table shows the calculations:
X | Y | |Y-X| | c|Y-X| |
-3 | -2 | 1 | 1c |
-3 | -1 | 2 | 2c |
-3 | 0 | 3 | 3c |
-3 | 1 | 4 | 4c |
-1 | -2 | 1 | 1c |
-1 | -1 | 0 | 0 |
-1 | 0 | 1 | 1c |
-1 | 1 | 2 | 2c |
1 | -2 | 3 | 3c |
1 | -1 | 2 | 2c |
1 | 0 | 1 | 1c |
1 | 1 | 0 | 0 |
2 | -2 | 4 | 4c |
2 | -1 | 3 | 3c |
2 | 0 | 2 | 2c |
2 | 1 | 1 | 1c |
Total | 30c |
For a valid probability distribution 30c must be equal to 1 so
30c = 1
c = 1/30
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Following are the set of values for which X <= Y :
X | Y | |Y-X| | c|Y-X| |
-3 | -2 | 1 | 1c |
-3 | -1 | 2 | 2c |
-3 | 0 | 3 | 3c |
-3 | 1 | 4 | 4c |
-1 | -1 | 0 | 0 |
-1 | 0 | 1 | 1c |
-1 | 1 | 2 | 2c |
1 | 1 | 0 | 0 |
Total | 13c |
Therefore,
2. Suppose that X and Y have a discrete joint distribution for which the joint p.f....
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