  We consider the problem of randomly choosing the sensors of a linear
time-invariant dynamical system subject to process and measurement noise. We
sample the sensors independently and from the same distribution. We measure the
performance of a Kalman filter by its estimation error covariance. Using tools
from random matrix theory, we derive probabilistic bounds on the estimation
error covariance in the semi-definite sense. We indirectly improve the
performance of our Kalman filter for the maximum eigenvalue metric and show
that under certain conditions the optimal sampling distribution that minimizes
the maximum eigenvalue of the upper bound is the solution to an appropriately
defined convex optimization problem. Our numerical results show the efficacy of
the optimal sampling scheme in improving Kalman filter performance relative to
the trivial uniform sampling distribution and a greedy sampling $\textit{with
replacement}$ algorithm.
