The inverse probability integral transform is just the inverse of this : specifically, if Y has a uniform distribution on [ 0, 1 ] and if X has a cumulative distribution F _ X, then the random variable F _ X ^ {-1 } ( Y ) has the same distribution as X.
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This idea was further developed in a book published in 1837 " Grundz�ge der Wahrscheinlichkeitsrechnung mit besonderer Anwendung auf die Operationen der Feldme�kunst " ( Foundations of Probability Calculus with Special Application to the Operations of Land Surveying ) which applied probability theory and least squares techniques to construction and surveying and deduced an error law that was not based on inverse probability arguments.
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