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I saw this great video series from Khan Academy about the Poisson distribution. The video derives the Poisson distribution equation by using an example of counting the number of cars that pass by.

The equation for the Poisson distribution is:

$P(k $ events in interval$) = \frac{\lambda^k e^{-\lambda}}{k!}$

And $\lambda$ seems to only be dependent on the rate of observation: if I'm watching cars pass by on the road and I see a car going down the street once a minute, then the probability of seeing a car going down the street is just $1^1 e^{-1}/1! = 36.79\%$.

But now let's say that I know that the street I'm observing happens to be an on-ramp to a major highway. $\lambda$ then really becomes dependent on the time of day I'm observing the street. There's going to be a lot less traffic passing on that road at 3am than at 7am, for instance. It seems that if I include more information about this road, then my probability will change (and get better).

So I'm thinking - can we find some sort of $\lambda^*$ that reflects this fact?

  • What if $\lambda^* = \lambda(t)$? This now requires that I make observations for all times in the day, and my probability explicitly requires knowing the time of day. Not bad, but I'd like one equation
  • What if $\lambda^* = \bar{\lambda}?$ I still have to make observations at each time of the day, and then take an average. Problem is, if this on-ramp is only heavily used during rush hour then an average will unfairly skew $\lambda^*$ lower.
  • What if I weigh $\lambda^*$ by knowing how much traffic could have taken the on-ramp? For instance, if I both observe the number of cars that are in the right lane (for this example, consider that this lane lets you turn right OR go straight), then I could weigh each measurement - how many cars actually took the on-ramp divided by the total number of cars that could have, hour-by-hour.

Is there a good way to include that information into the Poisson distribution equation?

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    $\begingroup$ The treatment of Poisson processes with variable arrival rate $\lambda(t)$ is quite standard, but of course it is assumed that $\lambda(t)$ is a known function. But you not only want $\lambda(t)$ to be a function that needs to be estimated, you even refuse to assume that the value of $t$ is known. So, what meaning do you assign to the estimate that $\lambda(t)$ has value $5$, say, for some unknown value of time instant $t$, and has estimated value $3.9$, say, at some other unknown time instant $t^\prime$ ?? $\endgroup$ – Dilip Sarwate Dec 5 '16 at 16:53
  • $\begingroup$ Not that I don't want to acknowledge $t$, but for the model I'm considering I'm hoping for a "first cut" that is independent of time. If the resulting probability does not meet requirements then I can refine the model by explicitly knowing $t$. $\endgroup$ – Dang Khoa Dec 5 '16 at 17:06

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