Let's say there is an induction loop in a road capable of counting the number of cars passing over it. By keeping a list of moments for when a car passed the detection loop, I am able to determine the average interval $\mu$ and variance $\sigma^2$ there over. I also know the amount of intervals $N$ measured and the last moment a car passed the induction loop.
I wish to know the chance of no cars passing for a time interval since the last moment a car passed the loop. This question seems to ask for a Poisson-distribution, but these seem no to keep into account the variance of time intervals, nor do they seem to use the amount of measurements. Which method should I use to determine the probability of no car passing the detection loop within a certain interval?
- $\mu$: The average interval between two cars passing the induction loop
- $\sigma^2$: The variance in these intervals
- $N$: the amount of interval measurements
- $P(t,c)$: The chance of $c$ cars passing the induction loop in interval $t$
- $t$: the interval for which the probability is sought
- $c$: the amount of cars which should pass within the interval. 0 in my case.