This paper considers the dynamical behavior of a discrete-time random Riccati equation with Markovian binary switching arising from a Kalman filter with intermittent observations. A number of properties of the random Riccati equation are studied such as comparability, monotonicity, boundedness, weak convergence, and ergodicity. A stochastic order method is used to compare random Riccati equations and establish the monotonicity property in an expectation sense. By using the idea of stopping times, we present a sufficient condition for an expectation boundedness property. Moreover, the contraction and nonexpansion, the Riemannian metric, and the Markov-Feller operator are applied to the analysis of weak convergence and ergodicity properties. These properties allow us to establish a relation between time average and the limit of the expectation of covariance matrices. We also revisit the independently and identically distributed case. By making use of the comparison and boundedness results in this paper, we present a variant of the existence theorem about a critical value in existing literature.