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==Formal statement==
For any [[random variable]] <math>X\in\mathbb R</math>, the random variable <math>F_X^{-1}(U)</math> has the same law {{Clarify|date=May 2023}} as <math>X</math>, where <math>F_X^{-1}</math> is the [[Cumulative distribution function#Inverse_distribution_function_(quantile_function)|generalized inverse]] of the [[cumulative distribution function]] <math>F_X</math> of <math>X</math> and <math>U</math> is uniform on <math>[0,1]</math>.<ref name="mcneil2005">{{cite book | last1 = McNeil | first1 = Alexander J. | last2 = Frey | first2 = Rüdiger | last3 = Embrechts | first3 = Paul | title = Quantitative risk management | date=2005 | series=Princeton Series in Finance | publisher=Princeton University Press, Princeton, NJ | page=186 | isbn=0-691-12255-5}}</ref>
For [[Random_variable#Continuous_random_variable|continuous random variables]], the inverse probability integral transform is indeed the inverse of the [[probability integral transform]], which states that for a [[continuous random variable]] <math>X</math> with [[cumulative distribution function]] <math>F_X</math>, the random variable <math>U=F_X(X)</math> is [[uniform distribution (continuous)|uniform]] on <math>[0,1]</math>.
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