When learning sequential probability ratio test, I get the impression that one should know exactly what the hypothesis is, and what the likelihood function is, in order to calculate and accumulate the likelihood ratio.

But what if we don't know the exact form of likelihood function?

Suppose in a game, a person is faced with $N$ screens, and each screen will show one random number every second. The person is told that for each screen, all the random numbers will be generated from one specific probability distribution (and will never be changed), but exactly what distribution is not known. The person is also told that one of the distributions has an expectation $X$. The poor guy's job is to guess which screen has the distribution with expectation $X$ based on observing the numbers sequentially, and should decide AS QUICKLY AS possible once reaching enough confidence. What is the theoretically best way to finish this game?

I understand that central limit theorem (CLT) states that when observations are long enough, the sum of all random numbers from any distribution approaches normal distribution. But in this case the person may need to decide with only a few observations. And I am not sure whether CLT is applicable here.

Thanks for any hints and suggestions.


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