  The intrinsic luminosity functions of extremely fast jets found in many
active galaxies and gamma-ray bursts are difficult to measure since their
apparent luminosities are strongly affected by relativistic beaming. Past
studies have only provided analytical predictions for the beamed
characteristics of populations in which all jets have the same Lorentz factor.
However, the jets found in active galaxies are known to span a large range of
speeds. Here we derive analytical expressions for the expected Doppler factor
distributions and apparent (beamed) luminosity functions of randomly-oriented,
two-sided jet populations which have bulk Lorentz factors distributed according
to a simple power-law over a range [G1, G2]. We find that if a jet population
has a uniform or inverted Lorentz factor distribution, its Doppler factor
distribution and beamed luminosity function will be very similar to that of a
jet population with Lorentz factors all equal to G2. In particular, the beamed
and intrinsic luminosity functions will have the same slope at high
luminosities. In the case of a steeply declining Lorentz factor distribution,
the slopes at high luminosities will also be identical, but only up to an
apparent luminosity equivalent to the upper cutoff of the intrinsic luminosity
function. Also, at very high apparent luminosities, the slope will be
proportional to the power law slope of the Lorentz factor distribution. At very
low luminosities, the form of the apparent luminosity function is very
sensitive to the lower cutoff and steepness of the Lorentz factor distribution.
We discuss how it should be possible to recover useful information about the
intrinsic luminosity functions of relativistic jets by studying flux-limited
samples selected on the basis of beamed jet emission.
