In this technical note, event-based pulse-modulated control is studied for linear stochastic systems with an average quadratic performance over an infinite horizon. From the probabilistic representation of solutions to differential equations, an analytic expression of the quadratic performance is provided. For stable and critically stable first-order systems, the schemes for seeking optimal parameters are given. In the unstable case, the stabilization problem is discussed. Unstable systems (first-order or higher-order) with white noises cannot be stabilized with probability 1, except that the noises are known to be bounded.