[math-fun] Random integers
The concept of picking just one point at random from a probability distribution is not a rigorous mathematical concept, although maybe it could be made rigorous. Or maybe for some reason it cannot be. One bit of mathematical folklore is that there's no such thing as a random integer, if all integers are treated symmetrically. I have long felt skeptical of this claim. Here is one reason: Let ๐ผ denote an irrational number. For definiteness, say ๐ผ = sqrt(2). Let C = R/Z denote the (unit-circumference) circle group. Let g_n denote the element g_n = n*๐ผ (mod 1) in C. Then G = {g_n in C | n in Z} forms a countable dense subgroup of C. There is a well-defined group isomorphism J : G โ> Z taking g_n in G to the integer n in Z. The cosets C/G = {x + G | x in C} of G form an uncontable quotient group that is rarely if ever discussed among decent mathematicians. Let A denote a set consisting of exactly one element from each coset x + G. Then the subsets A_n = A + g_n of G form a partition of G by countably many sets that are rotated versions of each other. The mapping phi : C โ> Z defined as phi(x) = n where n is the unique integer such that x belongs to A_n. Finally, since it is generally accepted that one can select a point at random from the uniform distribution on the unit interval [0, 1), which we identify with C, suppose this is done and the point selected is x. Then a random integer is phi(x). By symmetry, all integers are treated equally. โDan
I like your idea. I think you are trying to transfer the uniform distribution on [0,1) over to Z. I am still trying to understand it. On Sun, 10 May 2020 at 19:29, Dan Asimov <dasimov@earthlink.net> wrote:
The concept of picking just one point at random from a probability distribution is not a rigorous mathematical concept, although maybe it could be made rigorous. Or maybe for some reason it cannot be.
Could you please explain further what you mean?
One bit of mathematical folklore is that there's no such thing as a random integer, if all integers are treated symmetrically.
i.e. a uniform probability distribution on the integers does not exist.
The cosets
C/G = {x + G | x in C}
of G form an uncontable quotient group that is rarely if ever discussed among decent mathematicians.
Let A denote a set consisting of exactly one element from each coset x + G.
Does the existence of A require the Axiom of Choice?
Then the subsets
A_n = A + g_n
of G form a partition of G by countably many sets that are rotated versions of each other.
subsets of C, did you mean?
Finally, since it is generally accepted that one can select a point at random from the uniform distribution on the unit interval [0, 1), which we identify with C, suppose this is done and the point selected is x.
i.e. The uniform probability distribution on [0,1) does exist (and the probability of each singleton subset {x} is 0). What would be the image of a subinterval, say [0,1/2], under phi? I can't see it. It should be half of Z, shouldn't it? I think it depends on the choice of A. Gary McGuire
I think the "symmetry" argument is destroyed by the non-constructive choice of A; i.e. I suspect there are some A which would assign non-zero probability to 37, some which select an even integer with probability other than 1/2 et cetera. On Tue, May 12, 2020 at 1:37 PM Gary McGuire <gary.mcguire@ucd.ie> wrote:
I like your idea. I think you are trying to transfer the uniform distribution on [0,1) over to Z. I am still trying to understand it.
On Sun, 10 May 2020 at 19:29, Dan Asimov <dasimov@earthlink.net> wrote:
The concept of picking just one point at random from a probability distribution is not a rigorous mathematical concept, although maybe it could be made rigorous. Or maybe for some reason it cannot be.
Could you please explain further what you mean?
One bit of mathematical folklore is that there's no such thing as a
random
integer, if all integers are treated symmetrically.
i.e. a uniform probability distribution on the integers does not exist.
The cosets
C/G = {x + G | x in C}
of G form an uncontable quotient group that is rarely if ever discussed among decent mathematicians.
Let A denote a set consisting of exactly one element from each coset x + G.
Does the existence of A require the Axiom of Choice?
Then the subsets
A_n = A + g_n
of G form a partition of G by countably many sets that are rotated versions of each other.
subsets of C, did you mean?
Finally, since it is generally accepted that one can select a point at random from the uniform distribution on the unit interval [0, 1), which
we
identify with C, suppose this is done and the point selected is x.
i.e. The uniform probability distribution on [0,1) does exist (and the probability of each singleton subset {x} is 0).
What would be the image of a subinterval, say [0,1/2], under phi? I can't see it. It should be half of Z, shouldn't it? I think it depends on the choice of A.
Gary McGuire _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
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Dan Asimov -
Gary McGuire -
Michael Collins