The number of people arrived during a time interval follows Poisson distribution. Suppose there is only one customer arriving in the shop during time $T$, how many people arrive during time $T/4$?
I think the time between two arrivers follows exponential distribution $\exp(1/T)$, so the expectation of "time between two arrivers" is $1/T$. Then during $T/4$ time, there are $T^2/4$ customers arrived. Am I correct?