I'm looking for the answers to the following questions: Consider a single dimensional Gaussian $\mathcal{N}(\mu,\sigma^2)$ where we wish to estimate $\displaystyle E_{z\sim\mathcal{N}(\mu,\sigma^2)}[f(z)]$ for some differentiable function $f$.
(a) Show that the score function estimator of a single sample is $\frac{f(z)(x-\mu)}{\sigma^4}$.
(b) Show that the reparam estimator of a single sample is $f'(z)$.
(c) For each estimator show an example where it has lower variance than the other.
Additional info:
The reparamatrization trick - in order to make a discrete variable a continuous one we will add a white noise $\varepsilon\sim(0,1)$ to the discrete variable.
score function: $g(\theta) = \frac{\partial{log(f(\theta))}}{\partial{\theta}} $
If you can please explain in which cases we will prefer to use each of the techniques mentioned above.