Since Hinkley's original work on exact inference for a change in a sequence of random variables, many authors have proposed different methods based either on exact distributions or asymptotic approximations to test for and estimate a change point in such a sequence. Here we concentrate on the Poisson case. We propose estimators, tests and confidence intervals obtained by adapting and modifying Hinkley's and Worsley's work and we do an extensive study of the small sample properties of these proposed methods of inference, also comparing them to methods based on asymptotic approximations. The methods are applied to a neural train data set.