  The sectional curvature of a compact Riemannian manifold M can be seen as a
random variable on the Grassmann bundle of 2-planes in TM endowed with the
Fubini-Study volume density. In this article we calculate the moments of this
random variable by integrating suitable local Riemannian invariants and discuss
the distribution of the sectional curvature of Riemannian products. Moreover we
calculate the moments and the distribution of the sectional curvature for all
compact symmetric spaces of rank 1 explicitly and derive a formula for the
moments of general symmetric spaces. Interpolating the explicit values for the
moments obtained we prove a weak version of the Hitchin-Thorpe Inequality.
