I am trying to find the answer for the following question. I have two distributions of numbers between 1 and 8, as illustrated here:
I draw random samples from each distribution, which we can call "bucket" here, with random length, and I need to predict which "bucket" form the two has the high distribution, based on both samples.
I have tried a few calculation methods, but I am not sure if they are correct. I will briefly mention them based on the sample examples (I prepared some obvious ones):
Sample from bucket 1:
[2, 3, 5, 1, 1, 5, 2, 7, 6]
Sample from bucket 2:
[6, 3, 5, 7, 8, 7, 4, 5, 6, 2, 5]
First, I tried a common bayesian rule, which gives me the probability over each sample (starting from a 50/50 prior), like, sample 1 (0.13 high, 0.87 low), sample 2 (0.85 high, 0.15 low). But that doesn't give me a probability over the buckets.
I have then tried several approaches, and I think the most accurate results came from the stress-strength method, suggested in this thread Probability that random variable B is greater than random variable A. Using this method and soving for P(sample 1 < sample 2) I get the following results: Z value of 0.60, which translates to 0.7257 when converted to standard distribution, therefore my results is:
P(sample 1 < sample 2) = 1 - 0.7257, which is a 23% chance that sample 1 comes from the "low" bucket.
I would like to know if this approach would be appropriate for this problem, and if not if there are any suggestions of methods I could use to solve this? Thank you very much in advance.