# Poisson process question?

Customers arrive at a shop with Poisson process of rate 3 per hour.

What is the distribution of the amount of time the owner of the shop has to wait until the ﬁrst customer arrives?

I'm unclear as to what is the objective of the question. Is it to find the average time for each customer? Should the answer be E(T) = 1/3 = 20 minutes?

Thanks.

Suppose $X$ is the Poisson variable with mean $3$. Then, let $T$ be the time of first arrival.
$P(T\geq t)=P($No arrival in first $t$ hours$)=P(X(t)=0)=\dfrac{e^{-3t}(3t)^0}{0!}=e^{-3t}$ which illustrates that the interarrival time follows Exponential$(3)$.