How can I determine how often an event occurs based on collected data how long one has to wait for an occurence

Question

This is an experiment I can only observe, not design/change. I make the following observations: A police officer frequently monitors the same traffic location in the same manner. I see the officer arrive and I note the time. It then takes x minutes until an incident occurs, the officer chases that car, and the observation is over. At a later point, perhaps even another day, the officer returns, and the game starts again. I have now a series of observations, x1..xn [in minutes]. It is thus trivial to calculate the mean and say "on average, one has to wait ... minutes to witness an incident". However, the statement I would like to make is "on average, .. incidents occur per day". I seems that should be possible, but I don't know how, and I suspect that just the average of wait times does not equal how frequently an event occurs on average. Note: Key here is that the officer does not observe continually 24hrs. They arrive from time to time, and then it takes x minutes for them to witness something. (Also to keep it simple assume a "perfect" officer who is invisible, sees exactly each incident while present and chases every incident they see.)

I don't have the statistical vocabulary, but what is special about this experiment is that observation is not continuous (observe around the clock), but rather like a sample, and the duration of that sample is not independent (observe for say 15min every day), but ends with an observation.

Bonus question: The way I worded it, the officer always stays indefinitely until an incident occurs. Sometimes, however, the officer "gives up" and just leaves without having observed an incident, and I would have a separate list of values, wait-time and no occurence. I have a gut feeling including this would make it much harder, but should be included, since I have that information.

Answer 1

Such problems are very well modeled by Poisson random process. The classical case is how many minutes do you have to wait for a train to come to a station, given the already collected data on train arrivals.

You may want to look here for more information.

Answer 2

I think your problem is probably well-described as a Poisson process. You can describe the Poisson process in two ways: based on the number of events that occur in a certain time interval, or alternatively using waiting times $\Delta t$ between events.

If car accidents are following Poissonian statistics, and they probably are, then it doesn't matter when you start your clock, i.e. when the police arrive. However, if the presence of the police officer has an effect on the occurrence of accidents, then what you estimate won't be just the accidents per day, but the accidents per day given that a police officer is present (and these two probably differ).

The distribution of waiting times between events for a Poisson process is $P(\Delta t) = \nu \exp(-\nu\Delta t)$, where $\nu$ is the frequency of accidents, i.e. the average number of accidents per unit of time. If you want to give the average number in a day, just multiply by the corresponding factor. As I said, if the process is Poissonian it doesn't matter when you start your clock, the interval you measure will follow that distribution. Also, it doesn't matter if you observe just one accident and quit. If the value of $\nu$ is the same for every day (warning, it might not!) then you can pool $\Delta t$ from different days together. You could gather your waiting times and make a histogram, and check whether they follow an exponential distribution or not. If they do, you can get the frequency just as $\nu = 1/\langle \Delta t \rangle$.

The assumptions underlying the Poisson process are (just copied from Wikipedia):

1. The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
2. The average rate at which events occur is independent of any occurrences. For simplicity, this is usually assumed to be constant, but may in practice vary with time.
3. Two events cannot occur at exactly the same instant; instead, at each very small sub-interval exactly one event either occurs or does not occur.

It seems to me that your problem fulfills these conditions reasonably. Actually, the fact that you don't measure continuously in the same place helps fulfill these conditions, because if an accident occurred in a certain place in a certain time, then shortly after this accident it will be less likely that a new one occurs since some sort of signalization or whatever might be present. But if you stop the observation after the accident, you will probably get observations that better fulfill the conditions of independence between events.

Regarding your bonus question: this can be described by a zero-inflated Poisson process. In a Poisson process, it could occur that in a certain time interval you don't make any observation. A zero-inflated Poisson process includes another source of zero observations, which are due not just by chance but by another process that leads to no observations. You can see it here https://en.wikipedia.org/wiki/Zero-inflated_model#Zero-inflated_Poisson, and the reference therein. In this case, the expression for the frequency based on the observation of waiting times might not be easy to get analitically.