Conditional Monte Carlo

Replacing a simulated quantity $H$ by $E[H\mid \mathcal{G}]$ preserves its expectation and reduces variance by the law of total variance. This is Rao-Blackwellization. For a random sum, conditioning on the count and all but one summand replaces an event indicator by a cumulative distribution function evaluation. The resulting estimator needs that conditional expectation to be computable.