The Relation Between Transverse and Radial Velocity Distributions for
  Observations of an Isotropic Velocity Field
  We examine the case of a random isotropic velocity field, in which one of the
velocity components (the "radial" component, with magnitude $v_z$) can be
measured easily, while measurement of the velocity perpendicular to this
component (the "transverse" component, with magnitude $v_T$) is more difficult
and requires long-time monitoring. Particularly important examples are the
motion of galaxies at cosmological distances and the interpretation of Gaia
data on the proper motion of stars in globular clusters and dwarf galaxies. We
address two questions: what is the probability distribution of $v_T$ for a
given $v_z$, and for what choice of $v_z$ is the expected value of $v_T$
maximized? We show that, for a given $v_z$, the probability that $v_T$ exceeds
some value $v_0$ is $p(v_T \ge v_0 | v_z) = {p_z(\sqrt{v_0^2 +
v_z^2}})/{p_z(v_z)}$, where $p_z(v_z)$ is the probability distribution of
$v_z$. The expected value of $v_T$ is maximized by choosing $v_z$ as large as
possible whenever $\ln p_z(\sqrt{v_z})$ has a positive second derivative, and
by taking $v_z$ as small as possible when this second derivative is negative.
