For player $i$, competition $c$, and rival $r$, the fitted <Poisson generalized linear mixed model> is
$$
Y_{icr}\mid b_i\sim\operatorname{Poisson}(\mu_{icr}),
$$
$$
\log\mu_{icr}
=0.7287-0.8899\mathbf1_{\{c=B\}}
+1.4961\mathbf1_{\{c=C\}}
+0.4385\mathbf1_{\{r=S\}}+b_i,
\qquad b_i\sim N(0,0.1168).
$$
A negative-binomial model is commonly obtained by gamma mixing of a multiplicative Poisson mean and has negative-binomial marginals. Here the Gaussian random intercept gives a Poisson-lognormal mixture and correlates observations from the same player.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 218 / 3 / b / Solution

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A Poisson generalized linear mixed model gives a count response a conditional <Poisson distribution> whose logarithmic mean contains both fixed effects and latent random effects.

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