  The number of extant individuals within a lineage, as exemplified by counts
of species numbers across genera in a higher taxonomic category, is known to be
a highly skewed distribution. Because the sublineages (such as genera in a
clade) themselves follow a random birth process, deriving the distribution of
lineage sizes involves averaging the solutions to a birth and death process
over the distribution of time intervals separating the origin of the lineages.
In this paper, we show that the resulting distributions can be represented by
hypergeometric functions of the second kind. We also provide approximations of
these distributions up to the second order, and compare these results to the
asymptotic distributions and numerical approximations used in previous studies.
For two limiting cases, one with a relatively high rate of lineage origin, one
with a low rate, the cumulative probability densities and percentiles are
compared to show that the approximations are robust over a wide rane of
parameters. It is proposed that the probability density distributions of
lineage size may have a number of relevant applications to biological problems
such as the coalescence of genetic lineages and in predicting the number of
species in living and extinct higher taxa, as these systems are special
instances of the underlying process analyzed in this paper.
