Science > Physics > Independent/Dependent Phases 10: Re-interpretation of f(x,y) = 1/2
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
07 Oct 2005 11:15:22 AM |
| Object: |
Independent/Dependent Phases 10: Re-interpretation of f(x,y) = 1/2 |
From Osher Doctorow
Consider now Probable Correlation, defined by:
1) P(X<-->Y)(x,y) = F(x,y) + R(x,y)
where R(x,y) is the (engineering) reliability of continuous random
variables X, Y, that is R(x,y) = P(X > x, Y > y). We know that:
2) Dxy[P(X<-->Y)(x,y) = f(x,y) + Dxy[R(x,y)]
and from examples in recent threads we have Dxy[R(x,y)] often equal to
f(x,y).
Since Probable Correlation doesn't involve error, it is arguably a
two-way Probable Causation (recall that ordinarily causation implies
correlation but not vice versa because of error), which in turn
suggests that Dxy of this is of fundamental importance.
So examining f(x,y) = 1/2 leads me to notice that although PI works for
Rare Events and conditional probability works for non-Rare Events, we
don't exclude PI from an intermediate regime and we don't really know
whether conditional probability works for an intermediate regime - in
particular, for events with probability between .05 and .50 (= 1/2).
This suggests that in fact PI applies to events with probability below
..5 and that conditional probability applies to event with probability
above .5 rather than drawing the line at below .05 versus above .05.
We can still call events of probability < .05 "Very Rare Events" to
distinguish that particular case.
So the acceleration vs deceleration of the Universe occurring around
f(x, y) = 1/2 (see my Acceleration thread recently) is arguably
associated with a transition between Rare (probability less than .5)
and Non-Rare (probability greater than .5) events.
This is one of the basic reasons for the importance of 1/2 in these
scenarios. The univariate scenario with fX(x) = 1/2 was only studied
to explore what happens in simpler situations, although it gives
considerable insights.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Independent/Dependent Phases 10: Re-interpretation of f(x,y) = 1/2 |
07 Oct 2005 11:27:35 AM |
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From Osher Doctorow
When I say that "we don't really know whether conditional probability
works for events of probability below .50 and above .05", I mean to say
that although it is defined there and gives results (via FY|X=x(y/x) =
F(x,y)/FX(x) for FX(x) not 0), it is more "natural" to regard it as
applicable away from the region where probability < .05, which is away
from the left tail and arguably above .50 for symmetric distributions
(above the mean for these) and whatever the corresponding location of
the median is for asymmetric distributions.
As for f(x,y) and even fX(x), .50 (= 1/2) is now the "natural" choice
of the corresponding transition for the pdfs by "duality". Although
f(x, y) = 1/2 doesn't have the same interpretation as F(x, y) = 1/2,
f(x, y) acts as a "dual" to F(x, y) as I've argued earlier in this and
other recent threads.
Osher Doctorow
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