 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/discrete-mathematics-continuous-mathematics-and-probability/9/a/ii/solution

Independence gives the probability generating function of the sum as the product:

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

μ + λ

. Uniqueness of the coefficients of a power series therefore gives $X + Y \sim Poisson (μ + λ)$ . The product rule for these generating functions requires the stated independence.

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