Editing Inverse transform sampling (section)

== Examples ==
* As an example, suppose we have a random variable <math> U \sim \mathrm{Unif}(0,1)</math> and a [[cumulative distribution function]]
: <math>
\begin{align}
 F(x)=1-\exp(-\sqrt{x})
\end{align}
</math>
: In order to perform an inversion we want to solve for <math>F(F^{-1}(u))=u</math>
: <math>
\begin{align}
 F(F^{-1}(u))&=u \\
 1-\exp\left(-\sqrt{F^{-1}(u)}\right) &= u \\
 F^{-1}(u) &= (-\log(1-u))^2 \\
&= (\log(1-u))^2
\end{align}
</math>
: From here we would perform steps one, two and three.

* As another example, we use the [[exponential distribution]] with <math>F_X(x)=1-e^{-\lambda x}</math> for x ≥ 0 (and 0 otherwise). By solving y=F(x) we obtain the inverse function
: <math>x = F^{-1}(y) = -\frac{1}{\lambda}\ln(1-y).</math>
:It means that if we draw some <math>y_0</math>from a <math> U \sim \mathrm{Unif}(0,1)</math> and compute <math>x_0 = F_X^{-1}(y_0) = -\frac{1}{\lambda}\ln(1-y_0),</math> This <math>x_0</math> has exponential distribution.
: The idea is illustrated in the following graph:

: [[File:Inverse transformation method for exponential distribution.jpg|thumb|none|400px|Random numbers y<sub>i</sub> are generated from a uniform distribution between 0 and 1, i.e. Y ~ U(0, 1). They are sketched as colored points on the y-axis. Each of the points is mapped according to x=F<sup>−1</sup>(y), which is shown with gray arrows for two example points. In this example, we have used an exponential distribution. Hence, for x ≥ 0, the probability density is <math>\varrho_X(x) = \lambda e^{-\lambda \, x}</math> and the cumulative distribution function is <math>F(x) = 1 - e^{-\lambda \, x}</math>. Therefore, <math>x = F^{-1}(y) = - \frac{\ln(1-y)}{\lambda}</math>. We can see that using this method, many points end up close to 0 and only few points end up having high x-values - just as it is expected for an exponential distribution.]]
: Note that the distribution does not change if we start with 1-y instead of y. For computational purposes, it therefore suffices to generate random numbers y in [0, 1] and then simply calculate
: <math>x = F^{-1}(y) = -\frac{1}{\lambda}\ln(y).</math>