Consider the following Binomial generalized linear model for data $y_{1}, \ldots, y_{n}$, with logit link function. The data $y_{1}, \ldots, y_{n}$ are regarded as observed values of independent random variables $Y_{1}, \ldots, Y_{n}$, where

$$Y_{i} \sim \operatorname{Bin}\left(1, \mu_{i}\right), \quad \log \frac{\mu_{i}}{1-\mu_{i}}=\beta^{\top} x_{i}, \quad i=1, \ldots, n,$$

where $\beta$ is an unknown $p$-dimensional parameter, and where $x_{1}, \ldots, x_{n}$ are known $p$ dimensional explanatory variables. Write down the likelihood function for $y=\left(y_{1}, \ldots, y_{n}\right)$ under this model.

Show that the maximum likelihood estimate $\hat{\beta}$ satisfies an equation of the form $X^{\top} y=X^{\top} \hat{\mu}$, where $X$ is the $p \times n$ matrix with rows $x_{1}^{\top}, \ldots, x_{n}^{\top}$, and where $\hat{\mu}=$ $\left(\hat{\mu}_{1}, \ldots, \hat{\mu}_{n}\right)$, with $\hat{\mu}_{i}$ a function of $x_{i}$ and $\hat{\beta}$, which you should specify.

Define the deviance $D(y ; \hat{\mu})$ and find an explicit expression for $D(y ; \hat{\mu})$ in terms of $y$ and $\hat{\mu}$ in the case of the model above.