 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2021/discrete-mathematics-continuous-mathematics-and-probability/7/b/iii/solution

For

Z = 2 (X + Y) + 1

G_{Z} (s) = E [s^{2 (X + Y) + 1}] = s G_{X} (s^{2}) G_{Y} (s^{2}) = s exp ((λ + μ) (s^{2} - 1)) .

The initial factor

s

accounts for the shift by one; substituting

s^{2}

accounts for doubling the random variable. Its support consists of positive odd integers.

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