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Suppose I have two samples that can be described by an observable. Call it x. x can take on any value from 0 to infinity.

The distribution of values of x for sample 1 can be described by the normalized probability distribution f(x). The distribution of values of x for sample 2 can be described by the normalized probability distribution g(x).

If I make single independent measurements of x for both samples, how can I express the probability that the observed value of x from sample 1 exceeds the observed value of x from sample 2?

Nothing I've come up with gives me an answer that makes sense. My internal check is that if f(x) and g(x) are identical, then the probability should (I would think) approach 0.5.