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:

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

for each $y > 0$.

We have:

\(\ds \map \Pr {Y \le y}\)

\(=\)

\(\ds \map \Pr {k X \le y}\)

\(\ds \)

\(=\)

\(\ds \map \Pr {X \le \frac y k}\)

\(\ds \)

\(=\)

\(\ds 1 - \map \exp {-\frac y {k \beta} }\)

Definition of Exponential Distribution

$\blacksquare$