We present a method of approximating the unique invariant measure and associated Lyapunov exponents of a random dynamical system defined by the iid composition of a family of maps in situations where only one Lyapunov exponent is observed. As a corollary, our construction also provides a method of estimating the top Lyapunov exponent of an iid random matrix product. We develop rigorous numerical bounds for our invariant measure approximations and prove convergence of our Lyapunov exponent estimates to the true values. Comparisons between the invariant measure and Lyapunov exponents estimates generated by both our method and conventional random iteration are given.