 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2013/discrete-mathematics-probability-and-continuous-mathematics/8/c/i/solution

This numbered item specifies the model used below. A convenient sample space is the collection of

36

(s_{1}, s_{2}) \in {1, \dots, 6}^{2}

, each with probability

1/36

. The corresponding random variables are determined by the sum and difference. Inverting the two equations gives

S_{1} = \frac{X + Y}{2}, S_{2} = \frac{X - Y}{2} .

Thus a candidate pair

(X, Y) = (i, j)

is possible exactly when

(i + j) /2

and

(i - j) /2

are both integers between

1

and

6

. In particular,

i

and

j

must have the same parity. This inverse description will determine the joint probability mass function.

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