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Two key quantities for any model of buyer behavior in noncontractual settings are E[X(t)], the
expected number of transactions made by a randomly chosen individual in the interval (0,t], and
E[Y(t)|x,t_x,T], the expected number of transactions in the interval (T,T+t] for an individual
who made x transactions in the interval (0,T] with their last transaction occurring at t_x.
Fader et al. (2005) present expressions for these two quantities for the BG/NBD model.
While the equations are correct, the associated derivations are incorrect for a portion of the
parameter space. The purpose of this note is to present an alternative derivation of these two
quantities that holds for the entire parameter space. Similar derivation errors
occur for these quantities under the Pareto/NBD and BG/BB models, and we present derivations that hold for the entire parameter space.
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