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

By first computing the first and second derivatives of the log-likelihood for $\beta$, describe the algorithm you would use to find the maximum likelihood estimator $\hat{\beta}$. $[$ Hint: Recall that if $Z \sim \operatorname{Pois}(\mu)$ then

$$\mathbb{P}(Z=k)=\frac{\mu^{k} e^{-\mu}}{k !}$$

for $k \in\{0,1,2, \ldots\}$.]