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The arrival times of customers in a bakery can be modeled by a Poisson process (Nt)t≥0 with some rate λ > 0. On average, there are four arrivals per unit of time.

I want to ask about the question above what does it mean by "four arrivals per unit of time" and how can I get the rate(lamda) by just knowing the above information?

Thank you!

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    $\begingroup$ For example, suppose the time interval is 10 minutes and customers arrive randomly and independently. In each consecutive 10 minute interval, there will be an integer number of arriving customers. Maybe 3 or 1 or 8, etc. The mean arrival rate of 4 customers per time interval just means exactly that. $\endgroup$ – Ed V Apr 7 at 18:23
  • $\begingroup$ so how i get the lamda by that random variables? is there a mathematical way to get the rate lamda? $\endgroup$ – Jonathan Alfred Apr 7 at 18:59
  • $\begingroup$ If the true arrival rate was 4 customers per 10 minute interval, then lambda = (4 customers / 10 minute interval) times (10 minutes) = 4 customers. So lambda = 4 in the Poisson distribution. I think it is as simple as that, but there are lots of heavy hitters here, so I am sure that someone will correct me if I am wrong. $\endgroup$ – Ed V Apr 7 at 19:05
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Your question is a good one. It is somewhat vague to say that the number of customers in a bakery can be modeled by a Poisson Process. To understand what that means, imagine this exercise:

  1. Count the number of people that will arrive at the bakery in the next 5 minutes. Say you saw 3 customers in the first 5 minutes. Record that number. Now repeat the process. Over the next 5 minutes count how many customers arrive at the bakery. This time only 2 customers arrived. Over the following 5 minutes - 4 customers, in the 5 minutes after that - 2 customers, etc. Repeat this process for a total of say 1000 times.
  2. Group the 1000 observations by their values. You may have ended up with 100 of the 1000 5-minute intervals with a value of 0 (no customers arrived during those 5 minute intervals), 200 5-minute intervals with a value of 1, 300 5-minute intervals with a value of 2, 350 5-minute intervals with a value of 3, and 50 5-minute intervals with a value of 4 (4 customers arrived during each of these 5-minute intervals).
  3. Call X a random variable which will take as its value the number of people arriving in a 5-minute interval. In your case, the random variable takes only values 0-4. The fact that the problem says that the number of customers in this bakery can be modeled by a Poisson process means that if you calculated the relative frequency for each value of X, it should be relatively close to that obtained by the theoretical Poisson distribution function:

enter image description here

where lamda stands for the expected number of customers during any 5-minute interval (as Ed V pointed out); this is what you are given as the average number of arrivals per unit of time.

Let's look at a couple values and compare the relative frequency from our sample with the predicted probability by the Poisson distribution function.

X=0 In our sample, we saw observed 0 customers in 100 of 1000 5-minute intervals, hence with a relative frequency of 0.10. On the other hand, the Poisson distribution predicts that

enter image description here

we would see 0 customers during 1.83% of the 5-minute intervals.

X=1 In our sample, we saw observed 1 customer in 200 of 1000 5-minute intervals, hence with a relative frequency of 0.20. On the other hand, the Poisson distribution predicts that

enter image description here

we would see 1 customer during 7.32% of the 5-minute intervals.

Here is a picture from Wikipedia of what a theoretical Poisson distribution with a lamda=4 would look like. enter image description here

https://en.wikipedia.org/wiki/Poisson_distribution

In my contrived example, it doesn't look like the number of customers at the bakery is modeled well by the Poisson. What could be wrong (aside from the fact that these numbers were completely made up)?

The fact that the arrival of customers at a bakery can be modeled by a Poisson process says that there is some interval of time, (like the 5-minute long interval we used above), such that if we counted the number of customers arriving during those intervals, and calculated those relative frequencies, they would be close to those predicted by the Poisson distribution function. But notice that we are not told what that interval is. It could be 2.5 minutes, 0.7 minutes, or 0.05 seconds. This magic interval length has to be such that if we imagined dividing it into n non-overlapping sub-intervals, where n is large: a) the probability that 2 or more events (in our case the arrival of customers) occur in any given sub-interval is essentially 0, b) the events occur independently, and c) the probability that an event occurs during a given sub-interval is constant over the entire interval (dependence between intervals ok but not within intervals).

The Poisson Process most likely models well the arrival of customers in this bakery for only 1 magical interval length. For all other interval lengths, the Poisson Process most likely does not model it well, as some of the assumptions above (such as independence) would likely not hold. So to say that the Poisson process models something well without stating the interval length is to leave out a very interesting, crucial piece of information, that should prompt the student to wonder what interval might work in the problem at hand.

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    $\begingroup$ (+1) The point about the time interval is good. As an example, if an astronomer is using a small telescope to route photons to a detector, and the average rate of arrival of photons is 1 every 10 s, then they either need to have extended exposure time (increased photon collection time interval) or a bigger telesope aperture (increased average photon arrival rate) or both. Since the variance equals the mean, the signal-to-noise ratio would be poor if the product of the arrival rate and measurement time interval was short. $\endgroup$ – Ed V Apr 7 at 23:47
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    $\begingroup$ I love your example as it introduces an interesting layer of complexity. If we had an all-powerful telescope to see the minutest details then, as I understood it, we would find that a particular source would emit an average of n1 photons per an s second interval. And for some value of s, this process may be well approximated by the Poisson process. But since we only have a limited-capability telescope, we will only observe an average of n2<n1 photons per an s second interval, and thus we'll find that the Poisson model will model it well only if we increase the interval from s to t seconds. $\endgroup$ – ColorStatistics Apr 8 at 14:12
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    $\begingroup$ To view the bakery example via the "telescope lens" - In the bakery example, we do have the all powerful telescope so to speak because we see the true number of customers walking in. $\endgroup$ – ColorStatistics Apr 8 at 14:13
  • $\begingroup$ Exactly, for both your comments! Although I have to admit that the bakery example is always a little flakey to me! ;-) $\endgroup$ – Ed V Apr 9 at 0:06

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