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

Let

Y_{i}

be the first team's goal count in match

i

, let

x_{i}^{T}

be row

i

of the design matrix, and let

β

be the regression coefficient vector. The fitted Poisson regression assumes that the

Y_{i}

are independent random variables with

Y_{i} \sim Pois (μ_{i}), lo g μ_{i} = x_{i}^{T} β .

The likelihood function maximized by glm is therefore

L (β; y) = \prod_{i = 1}^{64} \frac{μ _{i}^{y_{i}} e ^{- μ_{i}}}{y _{i} !} = \prod_{i = 1}^{64} \frac{e x p { y _{i} x _{i}^{T} β - e ^{x_{i}^{T} β} }}{y _{i} !} .

Solved by gpt-5.6-sol high.

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