  In this paper we consider probabilistic analogues of some classical integral
geometric formulae: Weyl--Steiner tube formulae and the Chern--Federer
kinematic fundamental formula. The probabilistic building blocks are smooth,
real-valued random fields built up from i.i.d. copies of centered,
unit-variance smooth Gaussian fields on a manifold $M$. Specifically, we
consider random fields of the form $f_p=F(y_1(p),...,y_k(p))$ for $F\in
C^2(\mathbb{R}^k;\mathbb{R})$ and $(y_1,...,y_k)$ a vector of $C^2$ i.i.d.
centered, unit-variance Gaussian fields. The analogue of the Weyl--Steiner
formula for such Gaussian-related fields involves a power series expansion for
the Gaussian, rather than Lebesgue, volume of tubes: that is, power series
expansions related to the marginal distribution of the field $f$. The formal
expansions of the Gaussian volume of a tube are of independent geometric
interest. As in the classical Weyl--Steiner formulae, the coefficients in these
expansions show up in a kinematic formula for the expected Euler
characteristic, $\chi$, of the excursion sets $M\cap f^{-1}[u,+\infty)=M\cap
y^{-1}(F^{-1}[u,+\infty))$ of the field $f$. The motivation for studying the
expected Euler characteristic comes from the well-known approximation
$\mathbb{P}[\sup_{p\in M}f(p)\geq u]\simeq\mathbb{E}[\chi(f^{-1}[u,+\infty))]$.
