State and prove the Cramér-Rao inequality for a real-valued parameter $\theta$. [Necessary regularity conditions need not be stated.]

In a general decision problem, define what it means for a decision rule to be minimax.

Let $X_{1}, \ldots, X_{n}$ be i.i.d. from a $N(\theta, 1)$ distribution, where $\theta \in \Theta=[0, \infty)$. Prove carefully that $\bar{X}_{n}=\frac{1}{n} \sum_{i=1}^{n} X_{i}$ is minimax for quadratic risk on $\Theta$.