# Distribution for the number of sales per day / time between each sale

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?

• 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. Commented Jan 3, 2022 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 math.stackexchange.com/questions/4347815/…
– Basj
Commented Jan 4, 2022 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
Commented Jan 9, 2022 at 21:13