The properties of the telegrapher random process which is a Poissonian random walk on a straight line are studied in detail in probabilistic terms. The paper contains, besides the details of a rapid communication (Foong, 1992) by one of the authors, a number of new results. The distributions of the first passage time subject to an arbitrary number of reversals in the walk are obtained explicitly for both the starting directions. These distributions are then used to obtain, again explicitly, the corresponding distributions of the maximum of the walk, proving the conjecture by Orsingher (1990) for the one started moving right. The densities of the displacements from the origin in the presence of a trap are also given in detail. The relationship between this density and (1) the first passage time and (2) the maximum are given.
