Lyapunov exponents measure the rate of exponential expansion or contraction in a dynamical system. For a given nonlinear dynamical systems, it turns out to be a very hard problem to prove the nonvanishing of Lyapunov exponents or even to estimate them quantitatively. For a given prototypical two-dimensional map, we introduce a small random perturbation to the map and give with relative ease the quantitative estimate of the Lyapunov exponents versus the size of the randomness. This is a joint work with A. Blumenthal and L.-S. Young.