  As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli
component. This observation provides a tool for the extension of results which
are known for Bernoulli random variables to arbitrary distributions. Two
applications are provided here: i. an anti-concentration bound for a class of
functions of independent random variables, where probabilistic bounds are
extracted from combinatorial results, and ii. a proof, based on the Bernoulli
case, of spectral localization for random Schroedinger operators with arbitrary
probability distributions for the single site coupling constants. For a general
random variable, the Bernoulli component may be defined so that its conditional
variance is uniformly positive. The natural maximization problem is an optimal
transport question which is also addressed here.
