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Demographers conventionally estimate the mean number of children ever born to a woman who reaches a given age by adding age-specific fertility rates for her cohort up to that age. If the age selected is at the end of the childbearing period, the result is the cohort's total fertility rate (TFR); if only female babies are counted, the result is the gross reproduction rate (GRR). If births were generated by an age-dependent Poisson process, cumulative fertility rates give appropriate estimates, and so would the Nelson-Aalen estimator based on age-specific counts. Both of these methods are appropriate for that situation, but they estimate empirical mean numbers of births well, even though births are not generated by Poisson processes. This article shows why and when statistics like TFR and GRR are good estimates of mean counts, and how the same reasoning generalizes to the estimation of the mean number of any kind of event in any open population. [ABSTRACT FROM AUTHOR]