  [Abridged] Radio galaxies and quasars are among the largest and most powerful
single objects known and are believed to have had a significant impact on the
evolving Universe and its large scale structure. We explore the intrinsic and
extrinsic properties of the population of FRII objects (kinetic luminosities,
lifetimes, and the central densities of their environments). In particular, the
radio and kinetic luminosity functions of FRIIs are investigated using the
complete, flux limited radio catalogues of 3CRR and Best et al. We construct
multidimensional Monte Carlo simulations using semi-analytical models of FRII
radio source growth to create artificial samples of radio galaxies. Unlike
previous studies, we compare radio luminosity functions found with both the
observed and simulated data to explore the fundamental source parameters. We
allow the source physical properties to co-evolve with redshift, and we find
that all the investigated parameters most likely undergo cosmological
evolution. Strikingly, we find that the break in the kinetic luminosity
function must undergo redshift evolution of at least (1+z)^3. The fundamental
parameters are strongly degenerate, and independent constraints are necessary
to draw more precise conclusions. We use the estimated kinetic luminosity
functions to set constraints on the duty cycles of these powerful radio
sources. A comparison of the duty cycles of powerful FRIIs with those
determined from radiative luminosities of AGN of comparable black hole mass
suggests a transition in behaviour from high to low redshifts, corresponding to
either a drop in the typical black hole mass of powerful FRIIs at low
redshifts, or a transition to a kinetically-dominated, radiatively-inefficient
FRII population.
