  In active galactic nuclei, the relationship between UV and X-ray luminosity
is well studied (often characterised by $\alpha_\text{ox}$) but often with
heterogeneous samples. We have parametrized the intrinsic distribution of X-ray
luminosity, $L_\text{X}$, for the optically-selected sample of SDSS quasars in
the Stripe 82 and XXL fields across redshifts 0.5-3.5. We make use of the
available XMM observations and a custom pipeline to produce Bayesian
sensitivity curves that are used to derive the intrinsic X-ray distribution in
a hierarchical Bayesian framework. We find that the X-ray luminosity
distribution is well described by a Gaussian function in
${\log_{10}}L_\text{X}$ space with a mean that is dependent on the
monochromatic 2500A UV luminosity, $L_{2500}$. We also observe some redshift
dependence of the distribution. The mean of the $L_\text{X}$ distribution
increases with redshift while the width decreases. This weak but significant
redshift dependence leads to $L_{2500}$-$L_\text{X}$ and
$L_{2500}$-$\alpha_\text{ox}$ relations that evolve with redshift, and we
produce a redshift- and $L_{2500}$-dependent $\alpha_\text{ox}$ equation.
Neither black hole mass nor Eddington ratio appear to be potential drivers of
the redshift evolution.
