  We study the statistics of the number of connected components and the volume
of a random real algebraic hypersurface in RP^n defined by a Real Bombieri-Weyl
distributed homogeneous polynomial of degree d. We prove that the expectation
of the number of connected components of such hypersurface has order d^n, the
asymptotic being in d for n fixed. We do not restrict ourselves to the random
homogeneous case and we consider more generally random polynomials belonging to
a window of eigenspaces of the laplacian on the sphere S^n, proving that the
same asymptotic holds. As for the volume, we prove its expectation is of order
d. Both these behaviors exhibit expectation of maximal order in light of
Milnor's bound and the a priori bound for the volume.
