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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-4/13j/iii/solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 13J / iii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

Y_{i}

for team1_goal and

x_{i}

for team1_xg, the first hypothesis concerns only the expected value:

H_{1} : E (Y_{i} ∣ x_{i}) = x_{i} for every i .

(1)

The second specifies the entire conditional probability distribution:

H_{2} : Y_{i} ∣ x_{i} \sim Pois (x_{i}) for every i,

(2)

with independence between matches as assumed in the question. The Poisson limit theorem, often called the law of small numbers in this setting, suggests a Poisson distribution when a goal count is the sum of many approximately independent rare scoring opportunities.

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