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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 $x>\gamma$.

Rejection sampling is used because $g^*$ is exactly the conditional distribution of $X$ given $X>\gamma$. By drawing many samples from the original standard normal distribution and keeping only those that exceed the threshold $\gamma$, the retained samples are distributed according to $g^*$.

Conditions: Direct sampling from the original distribution $X \sim \mathrm{Normal}(0,1)$ is possible.; The acceptance rule is $X_i > \gamma$.

Rejection sampling is used because $g^*$ is exactly the conditional distribution of $X$ given $X>\gamma$. By drawing many samples from the original standard normal distribution and keeping only those that exceed the threshold $\gamma$, the retained samples are distributed according to $g^*$.

Conditions: Direct sampling from the original distribution $X \sim \mathrm{Normal}(0,1)$ is possible.; The acceptance rule is $X_i > \gamma$.