# Prove that $T(\textbf{X}) = \hat{\sigma}^{2}$ reaches the Cramer-Rao bound

Let $$X_{1},X_{2},\ldots,X_{n}$$ be a random sample whose distribution is given by $$\mathcal{N}(\mu,\sigma^{2})$$, where both parameters are unknown.

(a) Prove the normal probability density function satisfies the Cramer-Rao theorem hypothesis.

(b) Prove that $$T(\textbf{X}) = \hat{\sigma}^{2}$$ reaches the Cramer-Rao bound.

The exercise propose using the following result

RESULT

Suppose that $$X_{1},X_{2},\ldots,X_{n}$$ are IID whose joint probability density function is given by $$f(\textbf{x}|\theta)$$. If $$T(\textbf{X})$$ represents any unbiased estimator for $$\psi(\theta) = \textbf{E}_{\theta}(T(\textbf{X}))$$, then $$T(\textbf{X})$$ reach the Cramer-Rao bound if and only if \begin{align*} \frac{\partial\ln f(\textbf{x}|\theta)}{\partial\theta} = h(\theta)[T(\textbf{X}) - \psi(\theta)] \end{align*}

for some function $$h(\theta)$$.

I am really having trouble in proceeding with the exercise. Any help is appreciated. Thanks in advance!