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A simple proof of the probability integral transform theorem in probability and statistics is given that depends only on probabilistic concepts and elementary properties of continuous functions. This proof yields the theorem in its fullest generality. A similar theorem that forms the basis for the inverse method of random number generation is also discussed and contrasted to the probability integral transform theorem. Typical applications are discussed. Despite their generality and far reaching consequences, these theorems are remarkable in their simplicity and ease of proof.
John E. Angus (Thu,) studied this question.