# How many customers do I need to reach X purchases / second?

So I'm trying to answer this question:

Every customer has an average of a purchase every 10 minutes. That equals 1/600 purchases per second.

My question is: How many customers do I need in order to have a 1% chance of having 30 purchases per second?

I think that I should use poisson here, but kind of like 'reverse' poisson where I know the P(X) and I need to find out the mean. After that I just divide the mean by 1/600 and I get the number of customers. Is that approach correct?

• I'd say the problem is not well posed. First of all: what does average purchase mean? If a customer makes no purcheases during the year but then buys a lot of stuff on 31 dec, is it considered a purchase every 10 minutes? If so, the problem has no solution. Even if we consider that each customer has a rv $X$ which is "time till he purchases again", I doubt one can go from the mean of such a distribution to a precise probability – Ant Apr 11 '16 at 20:56

## 1 Answer

Hints:

For each number of customers, you'll have a different poisson distribution. The rate of that distribution will be proportional to the number of customers.

Solve the following equation for N:

$P(Poisson(N/600) > 30) > 1\%$

• Yup, this worked. Thanks a lot =) – Gaspa79 Apr 12 '16 at 16:54