= Solution
In a regular one-parameter model with joint density $f_\theta(x)$, write the <score function> as $U_\theta=\partial_\theta\log f_\theta(X)$ and the <Fisher information> as $I_n(\theta)=\mathbb E_\theta[U_\theta^2]$. Assume a parameter-independent support, differentiability and sufficient domination to interchange parameter differentiation and integration, and $0<I_n(\theta)<\infty$. If $T$ has finite <variance> and is unbiased for a differentiable target $g(\theta)$, the <Cramér-Rao lower bound> is
$$
\boxed{\operatorname{Var}_\theta(T)\geq\frac{[g'(\theta)]^2}{I_n(\theta)}}.
$$
Indeed, differentiating total mass gives $\mathbb E_\theta U_\theta=0$, and differentiating $\mathbb E_\theta T=g(\theta)$ gives $\operatorname{Cov}_\theta(T,U_\theta)=g'(\theta)$. <Cauchy-Schwarz inequality> proves the bound. For $n$ independent identically distributed observations, information adds: $I_n=nI_1$. In particular for the parameter itself, $g(\theta)=\theta$, the bound is $1/I_n$. The assumptions are material: the same formula cannot simply be imposed on models with parameter-dependent support.
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