Question

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

Answer 1

`X` and
`Y` are independent and identically distributed with PMF

`\mathrm{Pr}(X = x) = \begin{cases}1/6 & \text{if } x \in\{1,2,3,4,5,6\} \\ 0 & \text{otherwise}\end{cases}`

If
`Z=X-Y`, then
`Z` has range/support

{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

where we can get

• -5 = 1 - 6 in 1 way, hence with probability 1/36
• -4 = 1 - 5 = 2 - 6 in 2 ways, with probability 2/36 = 1/18
• -3 = 1 - 4 = 2 - 5 = 3 - 6 in 3 ways, with probability 3/36 = 1/12

and so on, so that the PMF of
`Z` is

`\mathrm{Pr}(Z=z) = \begin{cases}1/36 & \text{if } z\in\{-5,5\} \\ 2/36 & \text{if } z\in\{-4,4\} \\ 3/36 & \text{if }z\in\{-3,3\} \\ 4/36 & \text{if } z\in\{-2,2\} \\ 5/36 & \text{if } z\in\{-1,1\} \\ 6/36 & \text{if } z =0 \\ 0 & \text{otherwise}\end{cases}`