Uniform Distribution Misconception?
True or False?
It is impossible to choose a positive integer uniformly at random.
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1 solution
Just to play the devil's advocate, Comrade: Would your argument not apply to the real numbers on an interval as well?
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I'm not sure I understand your question correctly, but the probability of getting a real number on an interval is 0, because in this "continuous case" we assign probabilities to intervals (a single (real) number is interval of size 0, hence zero probability). The case with positive integers is the "discrete case," so my argument above won't be compatible to an interval of real numbers.
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Continuing my Devils' Advocate (or Socratic) dialogue: Why can't we assign probability 0 to the singletons in the case of the integers as well? Or what about the rational numbers or an interval of rational numbers?
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@Otto Bretscher – Well for the integers case, if we assign probability 0 to every singleton then the probability of getting any integer is 0, which is not true since it must be 1. That's what I mean by we can't get an integer uniformly at random. The case with rationals (in an interval or not) is the same as the case with the integers: they are countably infinite, so the same argument applies.
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@Huan Bui – Yes, exactly! The issue is not the discreteness of the integers but their countability; you might want to mention that in your solution. If your statement had been "It is impossible to choose a rational number on the interval [ 0 , 1 ] uniformly at random," I bet that more people would have gotten it wrong (but I understand that you wanted to start 2019 easy) .
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@Otto Bretscher – I agree. Well I hope the problem starts 2019 easy for most who attempt it.
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@Huan Bui – What about a problem with two parts: (I) It is possible to choose a real number on the interval [ 0 , 1 ] uniformly at random, and (II) It is possible to choose a rational number on the interval uniformly at random [ 0 , 1 ] , with four possible answers. It would teach people something about the standard axioms of probability (according to Comrade Kolmogorov).
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@Otto Bretscher – That's a good idea. Thanks, Otto!
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@Huan Bui – Now, I don't know enough about to "culture" of probability theory to judge how "standard" those axioms are. Are their competing systems of axioms? Does one have to state, explicitly, "according to Kolmogorov's axioms of probability"?
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@Otto Bretscher – I don't know enough about probability to answer this, but I learned Kolmogorov's axioms in MA381. I guess I should state that just to "stay safe."
Let us assume that we can indeed choose a positive integer uniformly at random. This means we assign to each positive integer a non-zero probability p of being selected. But this would imply the probability that any positive integer is selected is ∑ i = 1 ∞ p , which is not only greater than 1 but also diverges to infinity. So, True , it is impossible to choose a positive integer uniformly at random.