This paper investigates the remote state estimation problem over a bandwidth-limited channel. To improve the estimation quality, an event-based schedule is proposed to determine whether to transmit the current measurement to the remote estimator or not. We first show that it is erroneous to apply the widely used Gaussian-approximation assumption in the event-based scheduling. As a replacement, the generalized closed skew normal (GCSN) distribution is introduced to accurately portray the system state distribution in the event-based scheduling. Furthermore, we present the probability density functions of the minimum mean-squared error estimation algorithm exactly without approximation for the first time. The closed-form expressions for the mean and covariance of the GCSN distribution are derived as well. Numerical examples validate the effectiveness of the proposed algorithm.