Stochastic response of buildings isolated by lead–rubber bearings (LRB) is investigated. The earthquake excitation is modelled by non-stationary random process (i.e. uniformly modulated broadband excitation). The stochastic response of isolated building frames is obtained using the time-dependent equivalent linearization technique as the force–deformation behaviour of the LRB is highly non-linear. The non-stationary response of isolated structure is compared with the corresponding stationary response in order to study the influence of non-stationary characteristics of earthquake motion. For a given isolated building system and excitation, it is observed that there exists an optimum value of the yield strength of LRB for which the root mean square absolute acceleration of superstructure attains the minimum value. The optimum yield strength of LRB is obtained under important parametric variations such as isolation period and damping ratio of the LRB and the frequency content and intensity of earthquake excitation. It is shown that the above parameters have significant effects on the optimum yield strength of LRB. Finally, closed-form expressions for the optimum yield strength of LRB and corresponding response of the isolated structure are proposed. These expressions were derived based on the model of isolated structure with rigid superstructure condition subjected to stationary white-noise excitation. It was observed that there is a good comparison between the proposed closed-form expressions and actual optimum parameters and response of the isolated building system. These expressions can be used for initial optimal design of LRB for building system. Copyright © 2008 John Wiley & Sons, Ltd.