In this article, we first present the construction and basic properties of the Bochner integral for vector-valued functions on an arbitrary time scale. Using the properties of the Bochner integral, we develop an Lp-calculus for random processes on time scales, and present some results concerning the sample path and Lebesgue and Lp-integrability of a random process on time scales. Finally, we study random differential equations on time scales in the framework of the pth moment or Lp-calculus. An existence result is considered which gives sufficient conditions under which a sample path solution is also an Lp-solution.