Question

Let X and Y equal the outcomes when two fair six-sided dice are rolled. Let
W = X + Y. Assuming independence, find the p.m.f. of W when
(a) The first die has three faces numbered 0 and three faces numbered 2, and the second die has its faces numbered 0, 1, 4, 5, 8, and 9.
(b) The faces on the first die are numbered 0, 1, 2, 3, 4, and 5, and the faces on the second die are numbered 0, 6, 12, 18, 24, and 30.

Answer 1

Outcomes of the first die:
{0,0,0,2,2,2}
Outcomes of the second die:
{0,1,4,5,8,9}
There are 36 outcomes for W=x+y, some of which are identical. Make a table and compile the sums:
X 0 0 0 2 2 2
0 0 0 0 2 2 2
1 1 1 1 3 3 3
4 4 4 4 6 6 6
5 5 5 5 7 7 7
8 8 8 8 10 10 10
9 9 9 9 11 11 11

Now compile the count of the outcomes, and divide by 36 to get the pmf (probability mass function)
0 3 1/12
1 3 1/12
2 3 1/12
3 3 1/12
4 3 1/12
5 3 1/12
6 3 1/12
7 3 1/12
8 3 1/12
9 3 1/12
10 3 1/12
11 3 1/12

Part b is similar to part a.
Post your answer for a check if you wish.