Science > Physics > Quantum Gravity Via Expansion-Contraction 32.1: Back To Beta vs Raleigh
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Science > Physics |
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"OsherD" |
| Date: |
11 Nov 2006 12:19:56 AM |
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Quantum Gravity Via Expansion-Contraction 32.1: Back To Beta vs Raleigh |
From Osher Doctorow
The Rayleigh probability distribution function (pdf) can be looked up
in Wolfram or Wikipedia or Answers.com and so on. It has interesting
relationships to electricity, sound, and even stress/strain. It is
related to Raleigh Fading and Rician Fading in radio signals where
there is not or is a line of sight between transmitter and receiver
respectively (and see my Observer/Perception Sections of my last few
threads which relate to the latter two).
The Beta distribution is a finite interval distribution, whereas the
Raleigh pdf is a positive real line distribution. As I pointed out
recentlly in this thread, finite interval distributions are arguably
the best of the maximum PI entropy distributions since they are optimal
for two unknown parameters rather than 1 out of 2 unknown or both
parameters known. But almost equally important, the new concept of a
ratio of a variable and its derivative as being fundamental explains
the very odd form of the beta distribution.
The beta pdf is:
1) fX(x) = G(a1 + a2)x^(a1 - 1)(1 - x)^(a2 - 1)/(G(a1)G(a2)), 0 < x <
1, G gamma function
Now if we look at the factor:
2) x^(a1 - 1)/G(a1)
and similarly the other factor with x replaced by 1 - x and a1 replaced
by a2, we have in the beta pdf the product of two factors which are
variables divided by their derivatives when a1 and a2 are integers,
since G(a1) = (a1 - 1)! for a an integer, etc. The a1 - 1 st
derivative of x^(a1 - 1) for a1 a positive integer is (a1 - 1)!
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity Via Expansion-Contraction 32.1: Back To Beta vs Raleigh |
11 Nov 2006 12:31:24 AM |
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From Osher Doctorow
Schweinsberg's (2006) paper is only one of several new papers in Front
for the Mathematics ArXiv specifically on exponential functions in
probability.
Two others are "Hitting time of large subsets of the hypercube," by
Jiri Cerny and Veronique Gayrard of respectively Ecole Polytechnique
Federale De Lausanne Switzerland and CMI Marseilles France,
math.PR/0611242 v1 8 Nov 2006, and "Elementary potential theory on the
hypercube," by Gerard Ben Arous and Veronique Gayrard (Ben Arous is at
EPFL Lausanne Switzerland and NYU Courtant Institute New York),
math.PR/0611178 v1 7 Nov 2006.
I'll try to discuss these two papers more shortly.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity Via Expansion-Contraction 32.1: Back To Beta vs Raleigh |
11 Nov 2006 12:47:30 AM |
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From Osher Doctorow
The n-dimensional hypercube is {0, 1}^n. This hypercube is equipped
in Cerny and Gayrard (2006) with the metric:
1) d(x, y) = sum 1{x(i) not equal to y(i)}
with x(i) the coordinates of x in the hypercube, and Y_n or Yn the
simple random walk on the hypercube with transition probabilities 1/n
if d(x, y) is 1 and 0 otherwise.
Surprisingly, under quite general conditions, the hitting times of
large and possibly random sets are asymptotically exponential
distributed. The authors cite the Ben Arous and Gayrard paper (2006)
for some machinery. The latter paper is interesting for emphasizing
both the uniform pdf and the exponential pdf, about which they are
almost exclusively concerned. Potential Theory goes back quite far
and certainly isn't an "invention" of Ben Arous or Gayrard or Cerny.
Osher Doctorow
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