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 $\Exponential \cdot$ is the exponential distribution.

Proof

Let $Y \sim \Exponential \lambda$.

We aim to show that:

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

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

We have:

\(\ds \map \Pr {Y < -\beta \ln x}\)

\(=\)

\(\ds \frac 1 \beta \int_0^{-\beta \ln x} \map \exp {-\frac u \beta} \rd u\)

Definition of Exponential Distribution

\(\ds \)

\(=\)

\(\ds \intlimits {-\map \exp {-\frac u \beta} } 0 {-\beta \ln x}\)

Primitive of $\exp a x$

\(\ds \)

\(=\)

\(\ds -\map \exp {-\frac {-\beta \ln x} {\beta} } + \map \exp 0\)

\(\ds \)

\(=\)

\(\ds 1 - x\)

Definition of Natural Logarithm, Exponential of Zero

\(\ds \)

\(=\)

\(\ds \frac 1 {1 - 0} \int_x^1 \rd u\)

Primitive of Constant

\(\ds \)

\(=\)

\(\ds \map \Pr {X > x}\)

Definition of Continuous Uniform Distribution

$\blacksquare$