Consider a binomial generalised linear model for data $y_{1}, \ldots, y_{n}$, modelled as realisations of independent $Y_{i} \sim \operatorname{Bin}\left(1, \mu_{i}\right)$ and $\operatorname{logit} \operatorname{link}$, i.e. $\log \frac{\mu_{i}}{1-\mu_{i}}=\beta x_{i}$, for some known constants $x_{1}, \ldots, x_{n}$, and an unknown parameter $\beta$. Find the log-likelihood for $\beta$, and the likelihood equations that must be solved to find the maximum likelihood estimator $\hat{\beta}$ of $\beta$.

Compute the first and second derivatives of the log-likelihood for $\beta$, and explain the algorithm you would use to find $\hat{\beta}$.