Re: Random Variable on rationals in [0, 1]

On Feb 12, 11:43 pm, Sujit <sujit.gu...@xxxxxxxxx> wrote:
I have found F using first 200 rationals.
It wont be exact. But it should give pretty good
idea about graph of F as the probabilities which I ignored
would not sum up more than 2^(-200).

But this is not exact solution or some closed form solution.

You are right that your calculation of F is very close. Reaching a
closed form probably requires
a clever choice of bijection between the rationals and N. In your
bijection, it appears hard
to answer the question of "what is the 100th rational in the list?",
and this makes it hard to
find an explicit sum. There are simple bijections between N and N x
N, but the common factors
keep getting in the way. Maybe thinking about Farey sequences will
help. Alternately, you
could embrace the problem. Order the "rationals in (0,1)" as 1/2,
1/3, 2/3, 1/4, 2/4, 3/4, ...
where it is easy to figure out which one is the 100th, but a little
harder to figure out
the weight of 1/2, as it is 1/2 + 1/32 (for 2/4) + ...
.

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