Cumulative Distribution Function Inverse Transform
Result
This result is called sometimes called the probability inverse transform.
Let $(\Omega , \mathcal{F} , \mathbfsf{P} )$ be a
probability space and let $X: \Omega \to \R $
be a random variable with cumulative distribution
function $\rvcdf{X}: \R \to [0, 1]$.
Suppose $\inv{\rvcdf{X}}: [0, 1] \to \R $
exists, then $Y = \inv{\rvcdf{X}} \circ X$ is a
random variable with cumulative distribution
function $\rvcdf{Y}: [0, 1] \to [0, 1]$
satisfying $\rvcdf{Y}(y) = y$.
The conclusion is equivalent to the following:
$Y$ has a density and that density is the the
standard unform density (see \sheetref{uniform
densities}{Uniform Densities}).
Express $\rvcdf{Y}(\gamma ) = \mathbfsf{P} [Y \leq
\gamma ] = \mathbfsf{P} (\inv{Y}([0, \gamma ]))$
Notice1
\[ \begin{aligned} Y^{-1}([0, \gamma ]]) &= \Set{\omega \in \Omega }{Y(\omega ) \leq \gamma } \\ &= \Set{\omega \in \Omega }{\rvcdf{X}(X(\omega )) \leq \gamma } \\ &= \Set{\omega \in \Omega }{X(\omega ) \leq \rvcdf{X}^{-1}(\gamma )}. &= X^{-1}(\cdots). \end{aligned} \]
Using different notation the above can be
expressed succinctly as
\[ \begin{aligned} \rvcdf{Y}(\gamma ) = \mathbfsf{P} [Y \leq \gamma ] &= \mathbfsf{P} [\rvcdf{X} \circ X \leq \gamma ] \\ &= \mathbfsf{P} [X \leq \inv{\rvcdf{X}}(\gamma )] = \rvcdf{X}(\inv{\rvcdf{X}}(\gamma )) = \gamma . \end{aligned} \]
Future editions will discuss inverse transform sampling.
- Future editions will complete. ↩︎