Suppose that a die is rolled twice. What are the possible values that the following random variables can take on: (a) the maximum value to appear in the two rolls; (b) the minimum value to appear in the two rolls; (c) the sum of the two rolls; (d) the value of the first roll minus the value of the second roll?

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Best Answer

Let us suppose that a die is rolled twice. The maximum of two rolls is also random variable (not so trivial). What are the possible values?
We assume that die consists of numbers (1, 2, 3, 4, 5, 6) therefore there are 36 combinations that can occur.
Since for every k in (1, 2, 3, 4, 5, 6) ordered pair (k, k) can occur, maximum value can possibly achieve values of
(1,2,3, 4,5, 6).
Let us observe the minimal value to appear in the two rolls. Since (k, k) can obviously occur in two rolls (where k & (1, 2,3, 4, 5, 6).) minimal value can take values of (1, 2, 3, 4,5, 6).
In each roll numbers from 1 up to 6 can occur, therefore it is obvious that smallest sum is 2 and the largest is 12. It is trivial to see that every number inbetween can occur. Therefore the answer is:
(2,3,4,5,6,7,8,9, 10,11, 12).
It remains to answer what are the possible values when we consider first roll minus the second roll. As we have mentioned before in every roll numbers from (1, 2, 3, 4, 5, 6) can occur. Therefore it is obvious that the wanted set of numbers is:
(-5,-4, -3, -2, -1,0,1,2, 3, 4,5).
Hence, we are done.
Result:
We have: (1, 2,3, 4, 5, 6) for the a, b part, set (2, 3, 4,5, 6, 7,8, 9, 10, 11, 12) for the c part and finally (-5, -4, -3, -2, -1,0, 1, 2,3, 4,5) for the d part.