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A new inequality between some functional of probability distribution functions is given. The inequality is based on strict convexity of a function used in functional definition. Equality sign in the inequality gives a characteristic property of some probability distributions. This fact together with special character of functional is used to construct free-of-distribution two sample tests. Key words: convex functions; probability distances; characterization of distributions; Cram\'{e}r - von Mises distance; statistical tests.

A New Convexity-Based Inequality, Characterization of Probability Distributions, and Some Free-of-Distribution Tests