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Let us define "fastness" as a characteristic of how fast/agressive a driver is. One aspect of this is the (maximum) time between a traffic light turning red and the driver deciding they have to stop. The assumption here is that a "slow" driver will have low (read: very negative, like minus 5 seconds) value because they start stopping as soon as they notice the orange traffic light. On the contrary, a "fast" driver will have a higher value, like -1 second or even a positive one like 3 seconds, being "a bit" illegal.

(Disclaimer: fast and slow, as words, tend to give a positive "taste" on fast rather than slow. I chose them because they have few letters, and I do not advocate dangerous driving!)

By using this "time from red light to stop" as a random variable X, let's assume a normal distribution of drivers with known parameters m(average) and v(variance).

If that is so, what is the probability distribution that, when being stopped at a red light yourself, the driver at the frontmost of the queue is X-fast?

I guess slow drivers are more probable to appear there because they stop more easily/frequently. The logic behind the solution is more important than formulas here, and that is why I chose to post this here instead of math.se .

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  • $\begingroup$ You mention a queue near the end. Are drivers one behind another in a line or are they in parallel lanes independent of each other? Does everyone notice the orange light, or just the brake lights in front? And are crashes permitted? $\endgroup$ – flinty May 13 at 15:00
  • $\begingroup$ Well, rule of thumb is to create as simple a model as possible. Regarding your questions, let's assume one lane, everyone has full and immediate light knowledge, no crashes, no intersection space (as soon as you pass the light you are on safe zone), and also everyone travels at same speed (so the fastness of the car in front of you does not affect your decision to continue...only your distance from the traffic lights). Feel free to make any other assumption to make calculations easier. $\endgroup$ – George Menoutis May 13 at 17:32
  • $\begingroup$ you may need to define the distribution of red lights. Otherwise if there are no red lights then every driver is the same. $\endgroup$ – Dmitry Kamenetsky May 13 at 23:54
  • $\begingroup$ @Dmitry Kamenetsky I don't understand, could you elaborate? All I know is some roads have traffic lights. $\endgroup$ – George Menoutis May 14 at 6:18
  • $\begingroup$ if there are no traffic lights then there is no distribution. $\endgroup$ – Dmitry Kamenetsky May 14 at 11:24

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