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In this paper, we extend the Constraint Programming (CP) based functional test generation framework with a novel concept of distribution constraints. The proposed extension is motivated by requirements arising in the functional validation field, when a validation engineer needs to stress an interesting architectural event following some special knowledge of design under test or a specific validation plan. In such cases there arises the need to generate a sequence of test instructions or a collection of tests according to user-given distribution requirements which specify desired occurrence frequencies for interesting events. The proposed extension raises the expressive power of the CP based framework and allows specifying distribution requirements on a collection of Constraint Satisfaction Problem (CSP) solutions. We formalize the notion of distribution requirements by defining the concept of distribution constraints. We present two versions of problem definition for CP with distribution constraints, both of which arise in the context of functional test generation. The paper presents algorithms to solve each of these two problems. One family of the proposed algorithms is based on CP, while the other one makes use of both CP and the linear programming (LP) technology. All of the proposed algorithms can be efficiently parallelized taking advantage of the multi core technology. Finally, we present experimental results to demonstrate the effectiveness of proposed algorithms with respect to performance and distribution accuracy.