Solved – Is the Intersection Area Between 2 PDFs a Probability

Say that we have two processes $a$ and $b$ that generate numbers. Then say that the random variables $A$ and $B$ take values in the set of numbers that those processes generate, respectively. So $f_A$ and $f_B$ are their probability density functions, respectively.

Now, suppose that we plot those PDFs $f_A$ and $f_B$, and find that the intersection area between them is $0.6$. In other words (as Aksakal and Tim put it):
$$int_x text{min}big(f_A(x), f_B(x)big),,text{d}x = 0.6$$

Is $0.6$ a probability? Is it the probability that the processes $a$ and $b$ might generate numbers that are statistically indistinguishable?

Or, do I need to normalize $0.6$ somehow in order to make it into a probability?

To further clarify:

  • The question is not:

    "What is the probability that the processes are the same?". In fact, we can assume that the processes are necessarily different.

  • The question is rather:

    "Suppose that a randomly sampled number, from either one of the processes, was handed to you, without telling you the process that generated the number. Then, you were asked to predict the process that generated that number. Is this intersection area, $0.6$, the probability that the number that is given to you is statistically uninformative to you with respect to the mother process that generated it?"

If I understood your question correctly, then this is what you're looking for: $$int_xmin [f_A(x),f_B(x)] dx$$

Is it the probability that the processes a and b might generate numbers that are statistically indistinguishable?

No. Only if this quantity is equal to 1, you'll get two processes statistically indistinguishable. They'll be like two realizations of the same process.

UPDATE Say you have 2 rows of 10 bins each. You fill the first row with 100 red balls, and the second row with 100 blue balls. Next, you look at the first two bins, and pick either red or blue balls, whichever is the fewest. Then you do the same with a next pair of bins, and so on until all 10 pairs are exhausted. How many balls out of total 200 you collected? The quantity that you're looking at answers this question.

Overall, this type of quantity seems like something you might encounter in game theory or stochastic programming and dynamic decision process

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