Exercise: Tail probability via importance sampling

Estimate $\mathbb{P}(Z > 4)$ for $Z \sim N(0, 1)$ using:

(a) Plain Monte Carlo with $N = 10^5$.

(b) Importance sampling with $\tilde Z \sim N(\mu, 1)$ for $\mu = 4$.

True value: $1 - \Phi(4) \approx 3.17 \times 10^{-5}$.

Tasks

1. Implement both estimators. For each, report the point estimate, 95% CI, and the ratio of the IS standard error to the plain SE.

2. Vary $\mu \in \{0, 2, 4, 5, 7\}$. Plot the IS standard error against $\mu$.

3. Optimum. From the optimal-$q$ formula $q^* \propto f \cdot p$ (where $f = \mathbb{1}_{\{Z > 4\}}, p = \phi$), what's $q^*$? Approximate it with a normal — what mean and variance?

4. Compare $\mu = 4$ (centre on barrier) with $\mu = 5$ (slight overshoot). Why does $\mu = 4$ work well?