{\rm if}~M_B \leq M_B^*(z) \\   
\nonumber k = k_1 & {\rm if}~M_B > M_B^*(z)\ .   \end{aligned}$$

The parameters $\alpha$ and $\beta$ correspond to the faint-end and bright-end slopes of the optical LF, respectively, and $M^*_B(z=2)$ is the magnitude of the break in the double power-law shape of the LF at $z=2$. The actual values adopted in the LDLE parameterization, reported in Tab. \[ldle\], are derived by a fit to the observed LF. Extinction by Galactic dust is taken into account using the reddening $E(B-V)$ as a function of position, calculated by @schlegel98.

[ccccccc]{} EdS & 9.8 & -26.3 & -1.45 & -3.76 & 3.33 & 0.37\
$\Lambda$ & 5.0 & -26.7 & -1.45 & -3.76 & 3.33 & 0.30\

[cccccccccc]{} (1.0,0.0) & 8.49 & 6.44–10.46 & 1.58 & 0.089 & 0.063 & 0.368 & 0.151–0.585 & $1.75\pm 0.51$\
(0.3,0.7) & 8.64 & 6.56–10.64 & 1.56 & 0.088 & 0.062 & 0.461 & 0.224–0.698 & $1.37\pm 0.35$\

This approach, though computationally expensive, avoids biases in the generation of random samples of QSOs. It reproduces the observed LF and the distribution of redshifts and apparent magnitudes. Fig. \[mbzflat\] shows the observed and random generated QSOs in the $(z,M_B)$ space in the case 