Future Gaia and Legacy Survey of Space and Time data releases, together with
    wide area spectroscopic surveys, will deliver large samples of resolved binary
    stars with phase space coordinates, albeit with low-cadence. Given an
    eccentricity law $f(\epsilon)$, we derive properties of (i) the velocity
    distribution $v/\sqrt{G M/r}$ normalised by the value for a circular orbit at
    the measured separation $r$; (ii) the astrometric acceleration distributions
    $a/\left(G M/r^2\right)$ again normalised to the circular orbit value. Our
    formulation yields analytic predictions for the full statistical distribution
    for some commonly used eccentricity laws, if the timescale of data-sampling is
    comparable to or exceeds the binary period. In particular, the velocity
    distribution for the linear eccentricity law is surprisingly simple. With
    Bayesian analysis, we suggest a method to infer the eccentricity distribution
    based on the measured velocity distribution.

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