 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability-resit/9/a/iii/solution

Let

X_{1}, X_{2}

k \geq 0

P (X_{1} + X_{2} = k) = j = 0 \sum k P (X_{1} = j) P (X_{2} = k - j) = \frac{e ^{- (λ + μ)}}{k !} j = 0 \sum k (j k) λ^{j} μ^{k - j} = e^{- (λ + μ)} \frac{( λ + μ ) ^{k}}{k !},

where the last equality is the binomial theorem. Thus $X_{1} + X_{2} \sim Poisson (λ + μ)$ .

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