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.
Best Answer
The interarrival time follows an exponential distribution with parameter equal to the Poisson parameter. The proof is as follows:
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$(3)$.