For an interval 0< a < b , let (,) be the number of arrivals in the time interval (,).
I read a lot of resources on Poisson process but I am still not sure. I just want to confirm that for example if the N(0,4) represents the number of arrivals between the time (0,4). Would the distribution of N(0,4) follow the Poisson() distribution? Or would it be another distribution? Thank you for your help.
Yes $N(0,4)$ usually represents the number of arrivals in the time interval $[0, 4]$. If the Poisson process has rate $\lambda$, then $N(0,4) \sim \text{Poisson}(4 \lambda)$. More generally, the number of arrivals in a time interval of length $t$ is $\text{Poisson}(\lambda t)$.