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tokens: Since the Monte Carlo simulations described in step (2) are time consuming, we first constrained the parameter space of interest by testing a wide range of (⟨ β ⟩, rms( β )) values on a coarse sampling of the ranges ⟨ β ⟩ = − 2.2, − 1.0 and rms( β ) = 0.1, 1.5. We then accurately constrained their best-fit and confidence range through the analysis of a 35 × 35 grid in the range ⟨ β ⟩ = − 2.0, −1.4 for both the Gaussian and log-normal distributions, and rms ( β ) = 0.4, 0.8 (for the Gaussian distribution) and rms( β ) = 0.4, 1.0 (log-normal). For each position in the grid and for each magnitude bin, we extracted 25 000 objects from the CB07 library and perturbed magnitudes in all bands as described in 3). We note that the estimate of L in step (4) requires the observed and simulated UV slopes distributions to be binned: the choice of the bin widths was tailored to ensure that the extended but poorly populated tails in the observed distributions are efficiently sampled by the simulations. As shown in Fig. 5 , different bin widths were thus set depending on the 1, 2, 3 σ ranges of the observed distribution so as to reduce numerical noise in our procedure. The best-fit parameters of the Gaussian and log-normal PDF( β ) in each magnitude bin are reported in Table 2 . Confidence regions for the PDF( β ) are shown in Fig. 6 , together with observed and best-fit distributions. It is also highlighted how the latter (blue lines) are reshaped by observational effects (red): only a full modelling approach such as presented in C12 and adopted in this work allows us to recover the intrinsic slope distributions from observations. Table 2. UV slope distribution: best-fit parameters and 68% c.l. uncertainties. Fig. 6. Simulation results for the Gaussian ( left panel ) and log-normal ( right panel ) fit in the two different magnitude bins. In the leftmost column , we show maximum likelihood contours at 68%, 95%, and 99% c.l. in green, blue, and red, respectively; the red cross indicates the position of the best fit parameters. The rightmost column shows the histogram of the observed data and the relevant best-fit distribution. The red continuous line is the intrinsic PDF( β ), which is transformed into the blue continuous line by observational effects. We find that L > L * objects at z ∼ 3 have typical, average UV slopes ∼ − 1.7. No significant variation with UV magnitude is found in the Gaussian case. The log-normal best-fit distributions point to a slightly redder average (∼ − 1.5) in the fainter bin, although with large correlated uncertainties between the parameters. The absence of a strong correlation between UV slope and magnitude is in agreement with what was found by C12 at z ∼ 4, but it disagress with the previous analysis of z ∼ 3 objects by Bouwens et al. (2009) . While it is not possible to determine whether the best assumption is the Gaussian or the log-normal shapes, we note that the latter does not seem to accurately recover objects in the (poorly