Let $Y_{1}, \ldots, Y_{n}$ be independent Poisson random variables with means $\mu_{1}, \ldots, \mu_{n}$, for $i=1, \ldots, n$, where $\log \left(\mu_{i}\right)=\beta x_{i}$, for some known constants $x_{i}$ and an unknown parameter $\beta$. Find the log-likelihood for $\beta$.

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