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 logit $\operatorname{link} \mu_{i}=e^{\beta x_{i}} /\left(1+e^{\beta x_{i}}\right)$ for some known constants $x_{i}, i=1, \ldots, n$, and unknown scalar parameter $\beta$. Find the log-likelihood for $\beta$, and the likelihood equation that must be solved to find the maximum likelihood estimator $\hat{\beta}$ of $\beta$. Compute the second derivative of the log-likelihood for $\beta$, and explain the algorithm you would use to find $\hat{\beta}$.