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

## TL;DR

This paper develops a unified probabilistic framework using perturbation formulas for Poisson processes to derive integral representations, identities, and extensions relevant to various probability distributions and geometric formulas.

## Contribution

It introduces a unified approach to derive integral representations and identities for distributions and geometric formulas using perturbation techniques for Poisson processes.

## Key findings

- Derived integral representations for several distributions.
- Obtained new integro-differential identities for stable densities.
- Extended Crofton's derivative formula to Poisson processes.

## Abstract

The binomial, the negative binomial, the Poisson, the compound Poisson and the Erlang distribution do all admit integral representations with respect to its (continuous) parameter. We use the Margulis-Russo type formulas for Bernoulli and Poisson processes to derive these representations in a unified way and to provide a probabilistic interpretation for the derivatives. By similar variational methods, we obtain apparently new integro-differential identities which the density of a strictly $\alpha$-stable multivariate density satisfies. Then, we extend Crofton's derivative formula known in integral geometry to the case of a Poisson process. Finally we use this extension to give a new probabilistic proof of a version of this formula for binomial point processes.