Editing Inverse transform sampling (section)

==The method==
[[File:Generalized inversion method.svg|thumb|360px|Schematic of the inverse transform sampling. The inverse function of <math>y=F_X(x)</math> can be defined by <math>F_X^{-1}(y)=\mathrm{inf}\{x| F_X(x)\geq y\}</math>.]]
[[File:Inverse Transform Sampling Example.gif|thumb|360px|right|An animation of how inverse transform sampling generates normally distributed random values from uniformly distributed random values]]
The problem that the inverse transform sampling method solves is as follows:

*Let <math>X</math> be a [[random variable]] whose distribution can be described by the [[cumulative distribution function]] <math>F_X</math>.
*We want to generate values of <math>X</math> which are distributed according to this distribution.

The inverse transform sampling method works as follows:
#[[pseudorandom number generator|Generate a random number]] <math>u</math> from the standard uniform distribution in the interval <math>[0,1]</math>, i.e. from <math>U \sim \mathrm{Unif}[0,1].</math>
#Find the [[cumulative distribution function#Inverse_distribution_function_(quantile_function)|generalized inverse]] of the desired CDF, i.e. <math>F_X^{-1}(u)</math>.
# Compute <math>X'(u)=F_X^{-1}(u)</math>. The computed random variable <math>X'(U)</math> has distribution <math>F_X</math> and thereby the same law as <math>X</math>.

Expressed differently, given a cumulative distribution function <math>F_X</math> and a uniform variable <math>U\in[0,1]</math>, the random variable <math>X = F_X^{-1}(U)</math> has the distribution <math>F_X</math>.<ref name="mcneil2005" />

In the continuous case, a treatment of such inverse functions as objects satisfying differential equations can be given.<ref>{{cite journal | last1 = Steinbrecher | first1 = György | last2 = Shaw | first2 = William T. | title = Quantile mechanics | journal = European Journal of Applied Mathematics | date = 19 March 2008 | volume = 19 | issue = 2 | doi = 10.1017/S0956792508007341| s2cid = 6899308 }}</ref> Some such differential equations admit explicit [[power series]] solutions, despite their non-linearity.<ref>{{Cite journal |last1=Arridge |first1=Simon |last2=Maass |first2=Peter |last3=Öktem |first3=Ozan |last4=Schönlieb |first4=Carola-Bibiane |title=Solving inverse problems using data-driven models |journal=Acta Numerica |year=2019 |language=en |volume=28 |pages=1–174 |doi=10.1017/S0962492919000059 |s2cid=197480023 |issn=0962-4929|doi-access=free }}</ref>