  Order statistics theory is applied in this paper to probabilistic robust
control theory to compute the minimum sample size needed to come up with a
reliable estimate of an uncertain quantity under continuity assumption of the
related probability distribution. Also, the concept of distribution-free
tolerance intervals is applied to estimate the range of an uncertain quantity
and extract the information about its distribution. To overcome the limitations
imposed by the continuity assumption in the existing order statistics theory,
we have derived a cumulative distribution function of the order statistics
without the continuity assumption and developed an inequality showing that this
distribution has an upper bound which equals to the corresponding distribution
when the continuity assumption is satisfied. By applying this inequality, we
investigate the minimum computational effort needed to come up with an reliable
estimate for the upper bound (or lower bound) and the range of a quantity. We
also give conditions, which are much weaker than the absolute continuity
assumption, for the existence of such minimum sample size. Furthermore, the
issue of making tradeoff between performance level and risk is addressed and a
guideline for making this kind of tradeoff is established. This guideline can
be applied in general without continuity assumption.
