  We study the probabilistic sampling of a random variable, in which the
variable is sampled only if it falls outside a given set, which is called the
silence set. This helps us to understand optimal event-based sampling for the
special case of IID random processes, and also to understand the design of a
sub-optimal scheme for other cases. We consider the design of this
probabilistic sampling for a scalar, log-concave random variable, to minimize
either the mean square estimation error, or the mean absolute estimation error.
We show that the optimal silence interval: (i) is essentially unique, and (ii)
is the limit of an iterative procedure of centering. Further we show through
numerical experiments that super-level intervals seem to be remarkably
near-optimal for mean square estimation. Finally we use the Gauss inequality
for scalar unimodal densities, to show that probabilistic sampling gives a mean
square distortion that is less than a third of the distortion incurred by
periodic sampling, if the average sampling rate is between 0.3 and 0.9 samples
per tick.
