The only method of finding distribution function exactly, which avoids the subjective choice, is a characterization theorem. A theorem is on a characterization of a distribution function if it concludes that a set of conditions is satisfied by distribution function and only by the distribution function. In this book a family of probability distribution have been characterized through conditional expectations of the difference of two order statistics, record values and generalized order statistics conditioned on non-adjacent ordered statistics. The last three chapters based on single and product moments and also recurrence relations of moments of generalized ordered statistics and dual generalized order statistics from a general class of distribution which contains Burr XII distribution, Weibull distribution, power function distribution, Pareto distribution and exponential distribution etc. as its particular case .