Exponential Distribution in terms of Continuous Uniform Distribution

Theorem
Let $X \sim \mathrm U \hointl 0 1$ where $\mathrm U \hointl 0 1$ is the continuous uniform distribution on $\hointl 0 1$.

Let $\beta$ be a positive real number.

Then:


 * $-\beta \ln X \sim \Exponential \lambda$

where $\operatorname {Exp}$ is the exponential distribution.

Proof
Let $Y \sim \Exponential \lambda$.

We aim to show that:


 * $\displaystyle \map \Pr {Y < -\beta \ln x} = \map \Pr {X > x}$

for all $x \in \hointl 0 1$.

We have: