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Digital control design is commonly constrained to time-triggered control systems with periodic sampling. The emergence of more and more complex and distributed systems urges the development of advanced triggering schemes that utilize communication, computation, and energy resources more efficiently. This technical note addresses the question whether certainty equivalence is optimal for an event-triggered control system with resource constraints. The problem setting is an extension of the stochastic linear quadratic system framework, where the joint design of the control law and the event-triggering law minimizing a common objective is considered. Three differing variants are studied that reflect the resource constraints: a penalty term to acquire the resource, a limitation on the number of resource acquisitions, and a constraint on the average number of resource acquisitions. By reformulating the underlying optimization problem, a characterization of the optimal control law is possible. This characterization shows that the certainty equivalence controller is optimal for all three optimization problems.