Exponential of Negative of Exponential Random Variable has Beta Distribution

Theorem
Let $\beta$ be a positive real number.

Let $X \sim \Exponential \beta$ where $\Exponential \beta$ is the exponential distribution with parameter $\beta$.

Then:


 * $\displaystyle e^{-X} \sim \BetaDist {\frac 1 \beta} 1$

Proof
Note that if:


 * $\displaystyle Y \sim \BetaDist {\frac 1 \beta} 1$

then the probability density function of $Y$, $f_Y$ is given by:

for each $y > 0$.

Let:


 * $Z = e^{-X}$

It suffices to show that $Z$ has the same probability density function as $Y$.

We have:

By Derivative of Power, the probability density function of $Z$, $f_Z$ is therefore given by:


 * $\displaystyle \map {f_Z} z = \frac 1 \beta z^{\frac 1 \beta - 1}$

for each $z > 0$.

So:


 * $f_Y = f_Z$