In physics, chemistry, and mathematics the process of Brownian motion is often identified with the Wiener process that has infinitesimal increments. Recently many models of Brownian motion with finite velocity have been intensively studied. We consider in this paper one of such models, namely, a generalization of the Goldstein - Kac process to the three-dimensional case with Erlang-2 and Maxwell - Boltzmann distributions of velocities alternations. Despite the importance of having a three-dimensional isotropic random model for the motion of Brownian particles, numerous research efforts did not lead to an expression for the probability of the distribution of the particle position, the motion of which is described by the three-dimensional telegraph process. In case where a particle carries out its movement along the directions determined by the vertices of a regular n + 1-hedron in n-dimensional space was studied in [13] and closed-form results for the distribution of the particle position were obtained. In this paper we obtain expressions for the distribution function of the norm of the vector that defines particle's position at renewal instants in semi-Markov cases of the Erlang-2 and Maxwell - Boltzmann distributions, and study its properties. By knowing this distribution, we can determine the distribution of particle position since the motion of the particle is isotropic i.e., the direction of its movement is uniform distributed on the unit sphere in <$E{ roman bold R} sup 3>. Our results may be useful in studying the properties of an ideal gas.

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