Indeed some stuff is missing... Poisson distributions model independently random events that have a given average rate. Its variable counts the number of events

Indeed some stuff is missing... Poisson distributions model independently random events that have a given average rate. Its variable counts the number of events occurring before t. Waiting time between events is an exponential distribution (derived by computing the probability that 0 event occur before t). Fun application: suppose you want to take a bus, on average they are independently 10mins apart (rate 1/10). On average when you arrive uniformly random, the next bus will arrive in 10mins (and likewise the last one was 10mins ago). This can be intuitively understood because you have more chances to arrive at a time when buses are further apart. The mean of the waiting time is 1/rate and not 1/(2*rate) like we could imagine.