We propose an event-triggered control scheme for discrete-time linear systems subject to Gaussian white noise disturbances. The event-conditions are given in terms of the deviation between the actual system state and the state of a nominal undisturbed system whose state is identical to the real system state at the event times. In order to ensure that the conditional distribution of the deviation between the two systems, under the condition that no event occurs, remains a normal distribution, we employ thresholds that are themselves random variables. This allows us to: (i) provide expressions for the probability mass function of the times between events and, in turn, arbitrarily select this function; (ii) synthesize controllers associated with the proposed transmissions scheduler that are optimal in terms of an average quadratic cost. In particular, these two properties allow us to show that our event-triggered scheme is consistent in the sense that it outperforms (in a quadratic cost sense) traditional periodic control for the same average transmission rate and does not generate transmissions in the absence of disturbances. We demonstrate the effectiveness of our scheme in a numerical example and describe a way to solve the non-convex optimization problem arising in the approach.