Population statistics of beamed sources-III: Intrinsic probability density functions in the time domain

In a companion paper we have constructed a new statistical model for blazar populations, which reproduces the apparent velocity and redshift distributions from the MOJAVE survey while assuming single power law distributions for the Lorentz factors and the unbeamed monochromatic radio luminosity. Treating two separate cases, one for the BL Lac objects (BL Lacs) and one for the Flat Spectrum Radio Quasars (FSRQs), we calculated the distribution of the timescale modulation factor $\Delta t'/\Delta t$ which quantifies the change in observed timescales compared to the rest-frame ones due to redshift and relativistic compression. We found that $\Delta t'/\Delta t$ follows an exponential distribution with a mean depending on the flux limit of the sample, for both classes. In this work we produce the mathematical formalism that allows us to use this information in order to uncover the underlining rest-frame probability density function (PDF) of observable/measurable timescales of blazar jets, by fitting their observed probability density function. We test our proposed methodology extensively using simulated data, and identify limits of applicability and potential biases due to observational systematics and sample selection.