Re: Quantum Gravity-Dark Energy as Zero-Infinity (Coded as 0-1) Duals 2: Quantum/Macroscopic Erasure
- From: "OsherD" <mdoctorow@xxxxxxxxxxx>
- Date: 27 Feb 2006 21:17:54 -0800
From Osher Doctorow mdoctorow@xxxxxxxxxxx
The Dependent Probable Correlation between two continuous random
variables X, Y, a pointwise defined quantity, is thus a sum of two
major quantities excluding constant -1 (although -1 plays an important
role in making the result smaller in general than the Probable
Correlation itself). These quantities are the Probable Correlation
P(X<-->Y) = F(x,y) + P(X > x, Y > y) and the Logistic-related
expression which latter is also an inner product of forward and reverse
erasure type vectors.
In the expression x(1 - y) + y(1 - x) which is (x, y) * [(1, 1) - (y,
x)] with * meaning "dot product", the vector (1, 1) "comes from
x-infinity and y-infinity" when x is replaced by FX(x) and y by FY(y),
the (marginal) univariate cumulative distribution functions (cdfs) of X
and Y respectively. This is because FX(x) and FY(y) have maxima 1 and
in fact lim FX(x) = 1 as x --> infinity (or on [0, 1] as x --> 1- )
which turns out to be monotone nondecreasing FX(x).
On the other hand, the vector (x, y) can be regarded as "coming from
(0, 0) or (-infinity, -infinity) depending on whether X, Y are
repectively nonnegative real or all real in their supports. Moreover,
when x, y are very small and positive for X, Y nonnegative, we get Rare
Events. In terms of distributions, this involves the left tail of the
distribution (roughly speaking the left extreme of the "probability
graph" of the distribution), though (infinity, infinity) or its coding
as (1, 1) can also usually be regarded as a Rare Event involving the
right tail of the distribution. In terms of probability, the left tail
is a Rare Event-related tail for F(x,y) and the right tail is a Rare
Event-related tail for P(X > x, Y > y) since these probabilities are
very small (< .05) far out in the respective tails.
So not only are 0 and infinity or -infinity and +infinity involved in
DEPCOR, but Rare Events in respective left or right tails are
respectively involved in F(x, y) or P(X > x, Y > y) of Probable
Correlation P(X<-->Y)(x, y) = F(x,y) + P(X > x, Y > y).
Osher Doctorow
.
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