We study optimal filtering for continuous-time linear stochastic systems with Poisson-sampled observation processes. For each realization of the sampled observation process, the posterior distribution is a Gaussian process whose mean and covariance are described by continuous-discrete process. We are particularly interested in analyzing the expectation of the first and second moment of the estimation error with respect to the sampling process. Using the system-theoretic properties like observability and controllability, our results provide tractable conditions on the mean sampling rate for convergence of the expected error covariance, its boundedness and convergence of expected estimation error to zero. Some comparisons are also drawn with the solution of Riccati differential equation associated with the continuous-observation process.