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Let's say customers can buy one single product in a store open 24/7, and that the time of day does not influence anything (sales are equidistributed along the day).

With which classical distribution can we model the number of sales per day, and also the waiting time between two consecutive sales?

I would intuitively say a Poisson distribution for the former but I'm not sure about this, or if there are more specific distributions that can be used to model such situations.

Now if the sales are non equidistributed along the day, but instead like this:

Hour of day 00:00 to 06:00 06:00 to 12:00 12:00 to 18:00 18:00 to 24:00
Sales of the day 5% 35% 40% 20%

how can we model this?

kjetil b halvorsen
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Basj
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  • If the distribution is the same every day, my guess would be a normal distribution for the number of sales per day, follows from the CLT. The sales themselves are a Poisson point process (events occur continuously, independently, at a constant average rate). The wait time for Poisson point processes follows an exponential distribution. – Amaan M Jan 03 '22 at 22:09
  • @AmaanM Can you post your comment as an answer? I think your sentences #2 #3 are correct. For #1 I think it is a Poisson distribution, see the first comment of https://math.stackexchange.com/questions/4347815/link-between-poisson-distribution-and-exponential-distribution/4348044#comment9072705_4347815 – Basj Jan 04 '22 at 10:10
  • @AmaanM Applying the CLT requires that somehow you conceive of the number of sales as being the sum of a large number of identically distributed independent random variables. The number predicted by a homogeneous Poisson process will (of course) be a Poisson variable. If it has a large mean, it will be well approximated by a Normal distribution, but not otherwise. – whuber Jan 09 '22 at 21:13

1 Answers1

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The (time) points of sales can be modeled as a , and the simplest model is a Poisson point process. You also seem to be assuming a homogeneous process, that is, with an intensity function that do not vary with time. But the Poisson process assumes the events (sales) are independent, and, for example, the same customer making multiple buys close in time would violate that. So you should also look into more general point process models. See for instance Point Pattern Analysis- what is it good for.

But, if you have a Poisson point process, the time between events will have an exponential distribution. So a practical way of testing the Poisson assumption is to check if the waiting times do have an exponential distribution!

As for your additional question in comment, you have an inhomogeneous Poisson process, there is some information at

Eric Johnson
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kjetil b halvorsen
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  • Thanks! We assume a customer doesn't make multiple buys, so it seems correct. Something else: if the sales are not equidistributed along the day, but rather are centered in the middle of the day (and few sales at night), as described in the second part of the question, what kind of distribution law is this? – Basj Jan 09 '22 at 19:14