  In this paper a new version of the chain rule for calculating the mean square
derivative of a second-order stochastic process is proven. This random
operational calculus rule is applied to construct a rigorous mean square
solution of the random Chebyshev differential equation (r.C.d.e.) assuming mild
moment hypotheses on the random variables that appear as coefficients and
initial conditions of the corresponding initial value problem. Such solution is
represented through a mean square random power series. Moreover, reliable
approximations for the mean and standard deviation functions to the solution
stochastic process of the r.C.d.e. are given. Several examples, that illustrate
the theoretical results, are included.
