Using scale transformation on the exponential distribution

Question

In my textbook, they show that:

$$Y=\frac{X}{\lambda}\sim\text{Expo}(\lambda)$$

Where $X \sim\text{Expo}(1)$. I am confused about why they divide by $\lambda$ instead of multiply to transform from $\text{Expo}(1)$ to $\text{Expo}(\lambda)$. Since the exponential distribution has parameter $1$, wouldn't you want to multiply by $\lambda$?

Answer 1

You could easily multiply. If $c>0$, then the result given implies $$cX\sim\text{Exp}(1/c).$$

It seems like matter of style to me.

In general, we have that if $X\sim\text{Exp}(\lambda)$, and $\lambda, c>0$, then $$cX\sim\text{Exp}(\lambda/c).$$

Or as your book might put it, $$\frac{X}{c}\sim\text{Exp}(c\lambda).$$

Answer 2

Think of it this way: The parameter of the exponential distribution is the rate.

• $X$ is the time till the next Poisson point event occurring at rate $1$.

• $Y=X/\lambda$ is thus the (shorter for $\lambda>1$) time till the next Poisson point event occurring at (faster for $\lambda>1$) rate $\lambda$.