Russo's formula for Poisson point fields and its applications

We prove a new integral formula for Poisson point fields that is analogous to Russo's formula for Bernoulli fields. We use this formula to find the conditional distribution of the volume of the fundamental domain of the Voronoi mosaic that is generated by a homogeneous Poisson field of intensity $\lambda$ in $R^d$. Under the condition that the Voronoi polyhedron has $N$ hyperfaces, the volume of this domain has the gamma distribution $\Gamma (N,\lambda)$, which also provides an estimate for the distribution of the volume of a Voronoi polyhedron having $N$ hyperfaces.