I have run into some difficulty in understanding how poissonian rates are estimated from experimental data.
Consider the following scenario. I stand at the side of the road and start counting the number of cars driving past at time $t=0$. $N$ cars pass me by at times $T_{1}, T_{2}, ..., T_{N}$ where $T_{i} < T_{i+1}$. I stop counting the cars at time $T_{stop}$.
I see a number of different methods for estimating the poissonian rate for this situation:
- I divide the total time spent acquiring the data by the number of counts: $\lambda = N/T_{stop}$
- I consider the moment I count the first car as the "meaningful" start of this experiment and don't include the first car in the rate estimation: $\lambda = (N-1)/(T_{stop}-T_{1})$
- Building on the assumption in 2., I consider the time the final car passes as the "meaningful" end time for the experiment: $\lambda = (N-2)/(T_{N}-T_{1})$
Whilst a poissonian process is memoryless (an argument for method 1.), I believe that time intervals between consecutive poissonian events should be exponentially distributed. If this is the case, then isn't the definition of the time interval as $[0, T_{stop}]$ rather arbitrary? I don't think the intervals $T_{1}$ and $T_{stop}-T_{N}$ are exponential - they are completely defined by when I start and stop my stop watch, not by the process I am trying to measure.
What is the correct estimate for this process's arrival rate?