Final answer:
The distribution of inter-arrival times to a bank, when the number of customers is Poisson distributed, is exponential, not normal.
Step-by-step explanation:
The statement in the question is not correct. If a random variable, X, measuring the number of customers that enter a bank is assumed to be Poisson distributed, the distribution of inter-arrival times to that bank is not normal, but exponential.
The exponential distribution is often used to model the time between events (such as the arrival of customers) and it is a continuous distribution. It is the continuous analog of the Poisson distribution, which is a discrete distribution that models the number of events in a fixed interval of time or space.
So, in summary, the distribution of inter-arrival times to a bank, when the number of customers is Poisson distributed, is exponential, not normal.