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I have been reading the following motivation behind the Poisson distribution, and have been confused why we assume that every disjoint interval has the same probability of success based on the information we are given. Here is the passage:

To see a concrete example of how Poisson distribution arises, suppose we want to model the number of cell phone users initiating calls in a network during a time period, of duration (say) 1 minute. There are many customers in the network, and all of them can potentially make a call during this time period. However, only a very small fraction of them actually will. Under this scenario, it seems reasonable to make two assumptions:

  • The probability of having more than 1 customer initiating a call in any small time interval is negligible.
  • The initiation of calls in disjoint time intervals are independent events.

Then if we divide the one-minute time period into $n$ disjoint intervals, then the number of calls $X$ in that time period can be modeled as a binomial random variable with parameter $n$ and probability of success $p$, i.e., $p$ is the probability of having a call initiated in a time interval of length $1 \over n$.

I have 3 questions here:

  1. Why would we expect that each disjoint interval has the same probability $p$ of having a customer call in that interval?
  2. Is the reasoning behind expecting either a call or not in the time intervals because $n$ is very large, so we use our 2nd assumption above?
  3. What would the sample points in this scenario look like? Would they simply be collections of cell phone users in this network, and the times at which they call are properties which the people whom we sample possess, which depend on the way we have sampled the people? And we sample the people from the general population of all people? I am having trouble thinking exactly of what is sampled here, so I would appreciate some help with this.
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  • $\begingroup$ "Why would we expect that each disjoint interval has the same probability p of having a customer call in that interval?" In real life you wouldn't. So yeah the example is kind of a bad one. There are better real-life examples where Poisson is a good approximation. $\endgroup$ Nov 5, 2023 at 11:33

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For a fairly circular reason why we make that assumption, it's so that we wind up with a Poisson distribution rather than something else.

As for a practical argument, the assumption is that once we get to a small scale there isn't much real distinction between any of the individual intervals. For example, if our period of observation is between 12:00 and 12:01, and we take 10 intervals, then each interval is a 100ms slice of that minute. Is there any reason why we would expect that a phone call is more likely to happen between 12:00:00.300 and 12:00:00.400 than between 12:00:00.700 and 12:00:00.800? The answer is that outside of some very precise automated dialing system it seems like a fair approximation to say that the two intervals are equally likely to have a call happen in them.

Of course if the scale is bigger then that idea falls apart - if we look at the period between 12:00 and 18:00, then the probability that a phone call happens close to 12:00 is likely to be quite different to the probability of one happening closer to 18:00.

As for what we're sampling in this model - we are sampling the time intervals, and our variable of interest is $X_i = 1$ if a phone call happened in interval $i$, and $0$ if it didn't.

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  • $\begingroup$ How is the fact that in big intervals, the probability is different, not contradictory with how in tiny intervals, it is the same? It seems that the big intervals with greater probability, the tiny intervals which lie therein when dividing the space into tiny intervals would have greater probability than the rest of the tiny intervals $\endgroup$ Nov 6, 2023 at 0:15
  • $\begingroup$ The point is to choose a small enough time frame that the probabilities are approximately equal. No model is ever perfect, all that matters is that it's good enough. Is the actual probability of getting a call at 12:00:00 the same as the probability of getting one at 12:00:01? No, but is the difference going to be meaningful? Also no. Is there anything stopping you from assuming that it's the same as the probability of getting a call at 18:00:00? Nothing except the fact that if you try to use that to model a real phone network you'll probably find that it doesn't give you great results. $\endgroup$
    – ConMan
    Nov 6, 2023 at 0:19
  • $\begingroup$ What would be the intuition/justification behind why the smaller the intervals become, the more equal the probabilities become? $\endgroup$ Nov 6, 2023 at 0:21
  • $\begingroup$ It would be an assumption that the real probability is a smooth function of time, meaning that it can never vary more than some amount. If the intervals in question are small enough, then, that amount is going to be negligible, and in an appropriate limit it will tend to zero. $\endgroup$
    – ConMan
    Nov 6, 2023 at 0:25
  • $\begingroup$ Or to look at it another way, what's the argument against the assumption? Under what circumstances do you think that the probability of receiving a call at 12:00:00 is meaningfully different to the probability of receiving one at 12:00:01, and how does that change if we instead look at the probability of receiving a call at 12:00:00.1, or 12:00:00.01? (Assuming, of course, that time can be infinitely divided this way in the first place, but again this is just a mathematical model so it only has to be useful enough.) $\endgroup$
    – ConMan
    Nov 6, 2023 at 0:28

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