A fair die is rolled twice with the two rolls being independent of each other. Let M be the maximum of the two rolls and D be the value of the first roll minus the value of the second roll. Are M and D independent?

No, there will be a correlation between M and D. As one is large, the other will be large.

If M is 6, the difference can take on values of -5 to +5, with equal probability for each

If M is 1, the only allowed difference is 0. (snakeyes)

The mean of the D distribution (zero) is independent of M, but the standard deviation of D is not.

To determine if M and D are independent, we need to check if their joint probability distribution can be factored into the product of their marginal probability distributions.

Let's first find the joint probability distribution of M and D. Since the value of M is the maximum of the two rolls, we can represent it using two variables: X1 for the first roll and X2 for the second roll.

The possible values for X1 and X2 can range from 1 to 6, since we are rolling a fair six-sided die. The maximum of X1 and X2 will correspond to the maximum of the two rolls, which we'll denote as M.

To find the joint probability distribution, we need to determine the probability for each combination of X1 and X2.

Let's consider the cases for M:

1. If the maximum is 1, then both X1 and X2 must be 1.
2. If the maximum is 2, then either X1 is 2 and X2 is 1, or X1 is 1 and X2 is 2.
3. If the maximum is 3, then either X1 is 3 and X2 is either 1 or 2, or X1 is either 1 or 2 and X2 is 3.
4. If the maximum is 4, then either X1 is 4 and X2 is either 1, 2, or 3, or X1 is either 1, 2, or 3, and X2 is 4.
5. If the maximum is 5, then either X1 is 5 and X2 is either 1, 2, 3, or 4, or X1 is either 1, 2, 3, or 4, and X2 is 5.
6. If the maximum is 6, then X1 and X2 could be any combination of 1, 2, 3, 4, 5, and 6.

Now, let's consider the difference D:

D is the value of the first roll minus the value of the second roll. Since the rolls are independent, the difference can take any value from -5 to 5.

To determine the joint probability distribution of M and D, we need to find the probability for each combination of M and D. This involves considering all the possible values of X1 and X2 and calculating the probability for each combination.

Once we have the joint probability distribution, we can check if it can be factored into the product of the marginal probability distributions of M and D. If it can be factored, then M and D are independent. Otherwise, they are dependent.

I hope this explanation helps you understand the process of determining whether M and D are independent. To find the actual numbers, you can calculate the probabilities for each combination and perform the factorization check.