This paper investigates the optimal design of event-triggered estimation for linear systems. The synthesis approach is posed as a team decision problem where the decision makers are given by the event trigger and the estimator. The event-trigger decides upon its available measurements whether the estimator shall obtain the current state information by transmitting it through a resource constrained channel. The objective is to find the optimal tradeoff between the mean square estimation error and the expected number of transmissions over a finite horizon. After deriving basic characteristics of the optimal solution, we propose an iterative algorithm that alternates between optimizing one decision maker while fixing the other and vice versa. By analyzing the dynamical behavior of the iterative method, it is shown that the algorithm converges to a symmetric threshold policy for first-order systems if the statistics of the uncertainties are even and unimodal. In the case of bimodal distributions, we show numerically that the iterative method may find asymmetric threshold policies that outperform symmetric rules.