We consider the problem of Kalman filtering when observations are available according to a Bernoulli process. It is known that there exists a critical probability p. such that, if measurements are available with probability greater than p., then the expected prediction covariance is bounded for all initial conditions; otherwise, it is unbounded for some initial conditions. We show that, when the system observation matrix restricted to the observable subspace is invertible, the known lower bound on p. is, in fact, the exact critical probability. This result is based on a novel decomposition of positive semidefinite matrices.