Write down the model being fitted by the following $\mathrm{R}$ command, where $y \in\{0,1,2, \ldots\}^{n}$ and $X$ is an $n \times p$ matrix with real-valued entries.

fit $<-\operatorname{glm}(\mathrm{y} \sim \mathrm{X}, \mathrm{family}=$ poisson)

Write down the log-likelihood for the model. Explain why the command

$\operatorname{sum}(y)-\operatorname{sum}($ predict (fit, type $=$ "response" $))$

gives the answer 0, by arguing based on the log-likelihood you have written down. [Hint: Recall that if $Z \sim \operatorname{Pois}(\mu)$ then

$$\mathbb{P}(Z=k)=\frac{\mu^{k} e^{-\mu}}{k !}$$

for $k \in\{0,1,2, \ldots\}$.]