Re: Random Variable on rationals in [0, 1]
- From: Sujit <sujit.gujar@xxxxxxxxx>
- Date: Fri, 13 Feb 2009 02:43:56 EST
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.
---
Regards,
Sujit P Gujar
IISc Bangalore.
Web:http://clweb.csa.iisc.ernet.in/sujit
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
(p=0
---
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
function ofq=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
weX.
How to find out F(x) for x in [0,1]?
If we want F(x) to be correct up to some 2^{-i},
thanmay consider
first i+1 rationals and find out which are less
itor 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
rationalswill be
continuous everywhere except rationals in [0,1].
(Using Matlab)I have considered first 200
F.(according the order defined above) and plotted
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
- Follow-Ups:
- Re: Random Variable on rationals in [0, 1]
- From: Ross
- Re: Random Variable on rationals in [0, 1]
- References:
- Re: Random Variable on rationals in [0, 1]
- From: Ray Vickson
- Re: Random Variable on rationals in [0, 1]
- Prev by Date: Re: Testing limit = limit
- Next by Date: CAT TREE CHINA WWW.PETBED-CATTREE.COM CAT TREE FACTORY Boat Plane Car dog bed CHINA DOG BED China pet bed supplier WWW.PETBED-CATTREE.COM China pet products factory China Pet Dog bed Wrought iron dog car www.petbed-cattree.com china pet bed factory pet supplies pet supplier pet furniture pet product cat tree manufacturers
- Previous by thread: Re: Random Variable on rationals in [0, 1]
- Next by thread: Re: Random Variable on rationals in [0, 1]
- Index(es):
Relevant Pages
|
