Prove that under some regularity conditions such as the exchangeability of the integral and differentiation, we have

(a) \(E\left[\frac{1}{f\left(X_{i}, \boldsymbol{\theta}\right)} \frac{\partial}{\partial \boldsymbol{\theta}} f\left(X_{i}, \boldsymbol{\theta}\right)\right]=\mathbf{0}\). This shows that the score equation of the MLE is unbiased.

(b) \(\operatorname{Var}\left[\frac{1}{f\left(X_{i}, \boldsymbol{\theta}\right)} \frac{\partial}{\partial \boldsymbol{\theta}} f\left(X_{i}, \boldsymbol{\theta}\right)\right]=-E\left[\frac{\partial^{2}}{\partial \boldsymbol{\theta} \partial \boldsymbol{\theta}^{\top}} l(\boldsymbol{\theta})\right]\).