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The zero-variance density $g^*$ is the conditional distribution of the random variable $X$ given that the rare event $X>\gamma$ occurs. It is the ideal importance-sampling proposal that would make the estimator have zero variance.

Conditions: The target quantity is the rare-event probability $P(X>\gamma)$.; $g^*$ is supported only on the region $x>\gamma$.

The zero-variance density $g^*$ is the conditional distribution of the random variable $X$ given that the rare event $X>\gamma$ occurs. It is the ideal importance-sampling proposal that would make the estimator have zero variance.

Conditions: The target quantity is the rare-event probability $P(X>\gamma)$.; $g^*$ is supported only on the region $x>\gamma$.

The final estimator is the sample mean of the likelihood ratios multiplied by the rare-event indicator. It is computed by drawing samples from the fitted Normal proposal, calculating the ratio of the standard normal density to the proposal density, multiplying by the indicator that the sample exceeds $\gamma$, and averaging the results.

Conditions: Samples are drawn from the fitted Normal proposal, not the original distribution.; The event indicator is $\mathbf{1}_{\{Z_j>\gamma\}}$.

The final estimator is the sample mean of the likelihood ratios multiplied by the rare-event indicator. It is computed by drawing samples from the fitted Normal proposal, calculating the ratio of the standard normal density to the proposal density, multiplying by the indicator that the sample exceeds $\gamma$, and averaging the results.

Conditions: Samples are drawn from the fitted Normal proposal, not the original distribution.; The event indicator is $\mathbf{1}_{\{Z_j>\gamma\}}$.