From Osher Doctorow
It is well known in advanced statistics that if F(x, y) is joint
cumulative distribution function (cdf) of random variables X, Y, and if
FX(x), FY(y) are their univariate ("marginal") cdfs at x, y
respectively, then:
1) max{ FX(x) + FY(y) - 1, 0} < = F(x, y) < = min{FX(x), FY(y)}
For example, Kotz of Stanford and his colleagues have made use of this
in various papers and books.
So if FX(x) + FY(1) - 1 > = 0, then F(x, y) is related to them as
follows:
2) FX(x) + FY(y) - 1 < = F(x, y) (if left hand side is nonnegative)
The joint cdf F(x, y) and its marginals FX(x), FY(y) are fundamental to
conditional probability, Probable Influence/Causation (PI), and
Mainstream general probability-statistics regardless of explicit
conditionality or causation. So the appearance of an expression of
form:
3) FX(u) + FY(v) - 1 < = F(u, v) (if left hand side is nonnegative)
in all those branches of probability-statistics should be compared to
the three key expressions which have appeared in this thread:
4) 1 - x + y
5) x + y - xy
6) x - 1 + y
If we write x = FX(u), y = FY(v) in (3), then (3) can be written:
7) x + y - 1 < = F(u, v)
and rearranging terms:
8) x - 1 + y < = F(u, v)
So considerations of PI and conditional probability have led us to find
an underlying Mainstream probability-statistics cdf relationship to
trilinearity, bilinearity, Jacobson radicals (via star products or
circle composition products in (5) for the latter), etc., and thence to
physics and the minimal supersymmetric standard model, etc. Bilinear
and trilinear supersymmetry-breaking parameters A, B occur in the
minimal supersymmetric standard model. See for example John Ellis et
al (CERN, U. Minnesota) "Very constrained minimal supersymmetric
standard models," hep-ph/0405110 v1 12 May 2004.
Osher Doctorow
.
|
