Variables $Y_{1}, \ldots, Y_{n}$ are independent, with $Y_{i}$ having a density $p\left(y \mid \mu_{i}\right)$ governed by an unknown parameter $\mu_{i}$. Define the deviance for a model $M$ that imposes relationships between the $\left(\mu_{i}\right)$.

From this point on, suppose $Y_{i} \sim \operatorname{Poisson}\left(\mu_{i}\right)$. Write down the log-likelihood of data $y_{1}, \ldots, y_{n}$ as a function of $\mu_{1}, \ldots, \mu_{n}$.

Let $\widehat{\mu}_{i}$ be the maximum likelihood estimate of $\mu_{i}$ under model $M$. Show that the deviance for this model is given by

$$2 \sum_{i=1}^{n}\left\{y_{i} \log \frac{y_{i}}{\widehat{\mu}_{i}}-\left(y_{i}-\widehat{\mu}_{i}\right)\right\}$$

Now suppose that, under $M, \log \mu_{i}=\beta^{\mathrm{T}} x_{i}, i=1, \ldots, n$, where $x_{1}, \ldots, x_{n}$ are known $p$-dimensional explanatory variables and $\beta$ is an unknown $p$-dimensional parameter. Show that $\widehat{\mu}:=\left(\widehat{\mu}_{1}, \ldots, \widehat{\mu}_{n}\right)^{\mathrm{T}}$ satisfies $X^{\mathrm{T}} y=X^{\mathrm{T}} \widehat{\mu}$, where $y=\left(y_{1}, \ldots, y_{n}\right)^{\mathrm{T}}$ and $X$ is the $(n \times p)$ matrix with rows $x_{1}^{\mathrm{T}}, \ldots, x_{n}^{\mathrm{T}}$, and express this as an equation for the maximum likelihood estimate $\widehat{\beta}$ of $\beta$. [You are not required to solve this equation.]