I am observing 2 types of events with counts ($A$ and $B$) and I want to know if one is more frequent than the other for the time that I am observing. I know the chance of an appearance is uniform over time and independent of the other event. The observations are done for the same amount of time, at the same time.

Essentially I only have two numbers, e.g. $A=100$ and $B=85$.

Is there a test I can use to see if $A$ is more frequent than $B$?

  • $\begingroup$ Two data points is not something we can work with and have certainty in the result. Your observations could be due to chance, and so without more information or stronger assumptions on the data generating process, you need more data. $\endgroup$ – Demetri Pananos Sep 27 at 21:10
  • $\begingroup$ It sounds like you are observing two point processes. Depending on the assumptions you are willing to make, you could perform a two sample Poisson test to compare the intensity of the two processes $\endgroup$ – Breaking Waves Sep 28 at 2:51

If you are observing two point processes with constant rate that we can model as Poisson processes, for an equal time, the total counts for each process has a Poisson distribution, day $N_1, N_2$, which are independent. On the null hypothesis that the rates $\lambda_1, \lambda_2$ are equal, then, $N_1$ conditional on $N_1+N_1=n$ have a binomial distribution $\mathcal{Bin}(n, p=1/2)$, which can be used for a test.

Details can be found at Checking if two Poisson samples have the same mean

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