Given a 1-dimensional PDF with real numbers as output I need to find the average of distances $|a-b|$ between two outputs of such PDF. What I know is that the integral of the PDF must be 1 from -infinity to +infinity, and I somehow need to use an integral to sum up all the distances taking into account the probabilities of the outputs, but I don't know how.
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Can you compute the average difference between the numbers shown when you roll two dice? – John Hughes Nov 19 '17 at 19:04
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What if you have two unfair coins: each comes up heads 3/4 of the time. Call the difference "1" if one is heads, the other is tails, and the difference zero otherwise. Can you compute the average difference in that situation? – John Hughes Nov 19 '17 at 19:05
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1The average distance based on the PDF $f$ is $$\iint_{\mathbb R^2}|x-y|,f(x)f(y),dxdy$$ – Did Nov 19 '17 at 19:24
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@JohnHughes ?? The term "PDF" refers to continuous random variables. The results of coins and dice are discrete. – Did Nov 19 '17 at 19:27
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I was aware of that, Did. But if OP cannot answer the discrete question, the chances are very good that OP cannot answer the continuous one either; on the other hand, if the discrete question is easy, then the generalization to the continuous case (aside from questions of integrability) is relatively easy. Of course, it's even easier now that you've disclosed it, and OP can feel free to do the computation without ever understanding anything about the question at all. Forgive me for asking the kinds of leading questions from which I find students often learn. – John Hughes Nov 19 '17 at 19:32
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@Did Can it be generalized to two distributions? – Garmekain Nov 19 '17 at 19:37
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Yes. $ $ $ $ $ $ – Did Nov 19 '17 at 19:46