This paper studies the optimal output-feedback control of a linear time-invariant system where a stochastic event-based scheduler triggers the communication between the sensor and the controller. The primary goal of the use of this type of scheduling strategy is to provide significant reductions in the usage of the sensor-to-controller communication and, in turn, improve energy expenditure in the network. In this paper, we aim to design an admissible control policy, which is a function of the observed output, to minimize a quadratic cost function while employing a stochastic event-triggered scheduler that preserves the Gaussian property of the plant state and the estimation error. For the infinite horizon case, we present analytical expressions that quantify the tradeoff between the communication cost and control performance of such event-triggered control systems. This tradeoff is confirmed quantitatively via numerical examples. Besides, numerical simulations justify that the event-triggered control provides better quadratic control performance than the (traditional) periodic time-triggered control at the same average sampling rate.