Multiple of Exponential Random Variable has Exponential Distribution

Theorem
Let $\beta, k$ be real numbers with $\beta > 0$.

Let $X$ be a random variable.

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

Then:


 * $k X \sim \Exponential {k \beta}$

Proof
Let:


 * $Y \sim k X$

We aim to show that:


 * $\displaystyle \map \Pr {Y \le y} = 1 - \map \exp {-\frac y {k \beta} }$

for each $y > 0$.

We have: