  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.
