# Sum of many poisson random samples gives approximately normal distribution. But my events are still rare [closed]

It is well know that if $$X_1,X_2,.., X_n$$ are all independent and sampled from $$Poisson(\lambda)$$, then $$\sum_{i=1}^n{X_i}\sim Poisson(n\lambda)$$.

I have a following situation which is similar to the one here related discussion, however, I can not comment so I can not ask for a clarification on a specific term used in the discussion. The term is exposure. The situation is the following: I have an ab-test setup with two groups of users. Each about 3000 users. Then, in each group 45 and 53 cases are converted (the 0/1 event happened). I am treating these as rare events. Then we have poisson distribution for each group with $$\hat{\lambda}=\frac{events}{n}$$, where $$n$$ is number of , say, users per fixed time unit. Obviously, both lambdas will be small: (45/3000) and (53/3000), respectively. In the related discussion as well as in rate ratio test poisson this $$n$$ is called exposure. And now the question, if I follow the property of sum of poissons, I will get $$\hat{\lambda n}=\frac{events}{n}\times n=43$$ for group 1 and similarly for the other group. In this case a poisson with 43 events on average per unit of time is not rare anymore, but approximately normal. But in reality I know that 43 out of waiting 3000 "user-times" is rare. Isn't it a contradiction , is it? This exposure makes my individual rare count are now summed to a distribution with large lambda, not suitable for rare counts data.

Sorry for a bit of a messy question. May be I am missing some lambda scaling or other things.

## closed as unclear what you're asking by Tim♦Sep 11 at 15:03

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• I can't comprehend your question – Aksakal Sep 11 at 14:54