Inverse transform sampling: Difference between revisions

Revision as of 15:59, 25 March 2008

Revision as of 16:11, 25 March 2008

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:<math>F^{-1}(u) = \inf\;\{x \mid F(x)=u, 0<u<1\}</math>

''Claim:'' If ''U'' is a [[uniform distribution (continuous)|uniform]] random variable on <math>(0,&nbsp; 1)</math> then <math>F^{-1}(U)</math> follows the distribution ''F''.

''Proof:''

:<math>

:<math>\Pr(F^{-1}(U) \leq x) \!</math>

\begin{align}

:<math>= \Pr(\inf\;\{x \mid F(x)=U\} \leq x) \!</math> &nbsp;&nbsp; (by the definition of <math>F^{-1}</math>)

:<math>& \Pr(F^{-1}(U) \leq x) \!</math>\

:<math>= \Pr(U \leq F(x)) \!</math> &nbsp;&nbsp; (applying ''F'', which is [[monotonic function|monotonic]], to both sides)

:<math>& {} = \Pr(\inf\;\{x \mid F(x)=U\} \leq x) \!</math> &nbsp;&nbsp;quad \text{(by the definition of <math>}F^{-1}</math>) \\

:<math>= F(x) \!</math> &nbsp;&nbsp; (because <math>\Pr(U \leq y) = y</math>, since ''U'' is uniform on the unit interval)

:<math>& {} = \Pr(U \leq F(x)) \!</math> &nbsp;&nbsp;quad \text{(applying ''}F'',\text{ which is [[monotonic function|monotonic]], to both sides)} \\

:<math>& {} = F(x) \!</math> &nbsp;&nbsp;quad \text{(because <math>}\Pr(U \leq y) = y</math>,\text{ since ''}U''\text{ is uniform on the unit interval)}

\end{align}

</math>

== See also ==