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'''Inverse transform sampling''' (also known as '''inversion sampling''', the '''inverse probability integral transform''', the '''inverse transformation method''', or the '''[[Nikolai Smirnov (mathematician)|Smirnov]] transform''') is a basic method for [[pseudo-random number sampling]], i.e., for generating sample numbers at [[random]] from any [[probability distribution]] given its [[cumulative distribution function]].
Inverse transformation sampling takes [[Continuous uniform distribution|uniform samples]] of a number <math>u</math> between 0 and 1, interpreted as a probability, and then returns the smallest number <math>x\in\mathbb R</math> such that <math>F(x)\ge u</math> for the [[cumulative distribution function]] <math>F</math> of a random variable. For example, imagine that <math>F</math> is the standard [[normal distribution]] with mean zero and [[standard deviation]] one. The table below shows samples taken from the uniform distribution and their representation on the standard normal distribution.
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For any [[random variable]] <math>X\in\mathbb R</math>, the random variable <math>F_X^{-1}(U)</math> has the same distribution 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>.
[[File:InverseFunc.png|thumb|360px|Graph of the inversion technique from <math>x</math> to <math>F(x)</math>. On the bottom right we see the regular function and in the top left its inversion.]]
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