We suppose that a factory has on average 3 call per minutes. What is the probability to have 3 call in 4 minutes? I'm always confuse. Should I use a Poisson random variable or a stochastic process? i.e. if $X$ is the number of call in $4$ minutes, does $X\sim Poiss(4\lambda )$ or do I have to use a Poisson process $(X_t)$ s.t. $X_t\sim Poiss(\lambda t)$? Well, at the end, will have that if $Y\sim Poiss(4\lambda )$, then $$\mathbb P(X_4=3)=\mathbb P(Y=3),$$
but can someone explain me when I have to use a Poisson process or just a Poisson distribution?