 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-2/14/b/iii/solution

The zero-goal term contributes nothing to the expected value. Shift the summation index in the remaining terms:

μ = e^{- λ} k = 1 \sum \infty k \frac{λ ^{k}}{k !} = λ e^{- λ} k = 1 \sum \infty \frac{λ ^{k - 1}}{( k - 1 )!} = λ e^{- λ} e^{λ} = λ .

The exponential series justifies the convergent reindexing. Thus the parameter is the mean goals per game.

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