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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 |url=https://www.cambridge.org/core/journals/acta-numerica/article/solving-inverse-problems-using-datadriven-models/CE5B3725869AEAF46E04874115B0AB15 |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>
== Examples ==
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