Variance of exponential distribution when changing units

Question

I was tutoring a Stat student taking a probability course and I came across a problem dealing with variance of exponential that has left me confused.

$X \sim \mathrm{Exp}(\mathrm{rate}=\lambda)$ so $E(X)=1/\lambda$ and $\newcommand{\Var}{\mathrm{Var}}\Var(X)=1/\lambda^2$

Customers enter a store at a rate of 2 per hour, a customer has just arrived, what is the variability of the amount of time until the next customer arrives in minutes?

$\lambda=2, \; E(X)=1/2, \; \Var(X)=1/2^2=1/4$

Here the expected value = 30 minutes (1/2 of an hour) and the variance = 15 minutes (1/4 of an hour)

Now work on minutes scale:

$\lambda=2/60=1/30, \; E(X)=30, \; \Var(X)=30^2=900$

Here the expected value = 30 minutes but the variance = 900 minutes

I am confused why I am not getting the same variance when I switch from minutes to hour and vice versa.

Answer 1

Another way of looking at this is that if $X$ is the exponential variable in units of hours then $Y =cX$ converts $X$ to some other unit. $\newcommand{\Var}{\mathrm{Var}}\Var(Y) = c^2 \Var(X)$.

In your case $\Var(X)=1/4$ and $c=60$. So $\Var(Y)=3600$ and $\Var(X)=3600/4=900$.