Applications of the perturbation formula for Poisson processes to elementary and geometric probability

We present a unified approach to deriving integral representations for the binomial, negative binomial, Poisson, compound Poisson, and Erlang distributions with respect to their continuous parameters. This is achieved using Margulis-Russo-type formulas for Bernoulli and Poisson processes, which also provide a natural probabilistic interpretation of their derivatives. Extending these variational methods, we derive new integro-differential identities that characterize the densities of strictly α-stable multivariate distributions. We further generalize Crofton's derivative formula from integral geometry to the case of Poisson processes. This extension allows us to establish a new probabilistic proof of the formula for binomial point processes, highlighting the underlying geometric structure in a probabilistic framework.