  The joint probability distribution function (PDF) of the density within
multiple concentric spherical cells is considered. It is shown how its cumulant
generating function can be obtained at tree order in perturbation theory as the
Legendre transform of a function directly built in terms of the initial
moments. In the context of the upcoming generation of large-scale structure
surveys, it is conjectured that this result correctly models such a function
for finite values of the variance. Detailed consequences of this assumption are
explored. In particular the corresponding one-cell density probability
distribution at finite variance is computed for realistic power spectra, taking
into account its scale variation. It is found to be in agreement with
$\Lambda$-CDM simulations at the few percent level for a wide range of density
values and parameters. Related explicit analytic expansions at the low and high
density tails are given. The conditional (at fixed density) and marginal
probability of the slope -- the density difference between adjacent cells --
and its fluctuations is also computed from the two-cells joint PDF; it also
compares very well to simulations, in particular in under-dense regions, with a
significant reduced cosmic scatter compared to over-dense regions. It is
emphasized that this could prove useful when studying the statistical
properties of voids as it can serve as a statistical indicator to test gravity
models and/or probe key cosmological parameters.
