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

On Feb 12, 8:10 am, Sujit <sujit.gu...@xxxxxxxxx> wrote:
Is my question clear?

Yes, your question is clear, but the answer is not. To parphrase: you
construct a function f from the rationals in [0,1] to the integers
{1,2,3,...} and assign probability 2^(-i) to the i'th rational under
f. You then want the cumulative distribution F(x) back in [0,1]. You
claim to have found F for the first 200 rationals using Matlab, but I
don't believe it. Between x_i and x_j (1 <= i < j < 200) there will (I
suspect) be infinitely many other rationals that occur very far out in
the list, and just looking at the first 200 rationals won't pick up
all these intermediate points.

R.G. Vickson


---
Regards,
Sujit P Gujar
IISc Bangalore.
Web:http://clweb.csa.iisc.ernet.in/sujit

Hello

Let Omega = set of all rationals in [0, 1].
Now define random variable X on Omega.
As rationals are countable, we can order them.
We will define probability mass function(pmf) as
follows:
for i^{th} rational x,  f(X=x) = 2^(-i).

p/q be the representation of a rational st,
q != 0, gcd(p,q) = 1.
Now we can map each rational to integers with map
p/q |--> 2^p3^q

and for 1 ( p =1, q =1) map it to 2*3 and for 0 (p=0
q=1) to 3.

Now we order rationals according to the order
of their image under above map.
I mean is, we will say rational x is ith rational
in Omega if its image is i^{th} integer in wrt to
above map.
Now, Let F be the cumulative distribution function of
X.
How to find out F(x) for x in [0,1]?
If we want F(x) to be correct up to some 2^{-i}, we
may consider
first i+1 rationals and find out which are less than
or equal to x and sum their probabilities.

Is there any way to construct explicit formula to
find
F(x) in terms of x for the above random variable?
That is, put x in the formula and get F(x).

How in general the graph of F(x) will be? I guess it
will be
continuous everywhere except rationals in [0,1].

(Using Matlab)I have considered first 200 rationals
(according the order defined above) and plotted F.
I have attached the diagram for those who are
interested
in getting rough idea abt F.

---
Regards,
Sujit P Gujar
IISc Bangalore.
Web:http://clweb.csa.iisc.ernet.in/sujit



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