Exponential Random Numbers

The basic rule relating two continuous probability distributions $f(y)$ and $w(x)$ is

$$f(y) dy = w(x) dx$$

Let us assume a domain [0,1] in $y$ and $x$ and that the $y$ variable is uniformly distributed, $f(y) = 1$. Then

$$y(x) = \int_0^x dx' w(x')$$

with the condition

$$y(1) = 1 = \int_0^1 dx' w(x')$$

To get an exponential distribution set $w(x) = exp(-x)$ and extend the domain to $\infty$ to get the transformation

$$y(x) = 1 - exp( -x )$$

or

$$x = - ln( 1 - y )$$