State without proof the inequality known as the Cramér-Rao lower bound in a parametric model $\{f(\cdot, \theta): \theta \in \Theta\}, \Theta \subseteq \mathbb{R}$. Give an example of a maximum likelihood estimator that attains this lower bound, and justify your answer.

Give an example of a parametric model where the maximum likelihood estimator based on observations $X_{1}, \ldots, X_{n}$ is biased. State without proof an analogue of the Cramér-Rao inequality for biased estimators.

Define the concept of a minimax decision rule, and show that the maximum likelihood estimator $\hat{\theta}_{M L E}$ based on $X_{1}, \ldots, X_{n}$ in a $N(\theta, 1)$ model is minimax for estimating $\theta \in \Theta=\mathbb{R}$ in quadratic risk.