  By making use of the familiar Mathieu series and its generalizations, the
authors derive a number of new integral representations and present a
systematic study of probability density functions and probability distributions
associated with some generalizations of the Mathieu series. In particular, the
mathematical expectation, variance and the characteristic functions, related to
the probability density functions of the considered probability distributions
are derived. As a consequence, some interesting inequalities involving complete
monotonicity and log-convexity are derived.
