I roll two dice and observe two numbers X and Y. If Z = X − Y, find the range and PMF of Z.

The range of Z (X - Y) is from -5 to 5. This is because the smallest value X can be is 1 and the largest value Y can be is 6 (1 - 6 = -5), and the largest value X can be is 6 and the smallest value Y can be is 1 (6 - 1 = 5).

The PMF (probability mass function) of Z can be calculated as follows:

1. For Z = -5, -4, -3, -2, -1, the probabilities are 1/36, 2/36, 3/36, 4/36, 5/36 respectively. This is because there is only 1 way to get a difference of 5 (6 - 1), 2 ways to get a difference of 4 (6 - 2, 5 - 1), 3 ways to get a difference of 3 (6 - 3, 5 - 2, 4 - 1), 4 ways to get a difference of 2 (6 - 4, 5 - 3, 4 - 2, 3 - 1), and 5 ways to get a difference of 1 (6 - 5, 5 - 4, 4 - 3, 3 - 2, 2 - 1).

2. For Z = 0, the probability is 6/36. This is because there are 6 ways to get a difference of 0 (1 - 1, 2 - 2, 3 - 3, 4 - 4, 5 - 5, 6 - 6).

3. For Z = 1, 2, 3, 4, 5, the probabilities are 5/36, 4/36, 3/36, 2/36, 1/36 respectively. This is the same as the probabilities for Z = -1, -2, -3, -4, -5 but in reverse order, because the ways to get a positive difference are the same as the ways to get a negative difference, just with X and Y swapped.

So the PMF of Z is as follows:

P(Z = -5) = 1/36 P(Z = -4) = 2/36 P(Z = -3) = 3/36 P(Z = -2) = 4/36 P(Z = -1) = 5/36 P(Z = 0) = 6/36 P(Z = 1) = 5/36 P(Z = 2) = 4/36 P(Z = 3) = 3/36 P(Z = 4) = 2/36 P(Z = 5) = 1/36