From Osher Doctorow
I should have mentioned in the previous Section that we assumed that
p(x, y) was asymmetric and so we didn't have to include both FX and
FY.
In this Section, we'll looked for a second mixed partial derivative
equation of Probable Correlation and Entanglement which in the
Probable Causation/Influence (PI) terminology are given by:
1) P(A<-->B) = P(AB) + P(A ' B ' )
2) P(X<-->Y) = F(x, y) + P(X > x, Y > y)
where (A<-->B) is defined by:
3) (A<-->B) = (A-->B)(B-->A) = (A ' U B)(B ' U A) = AB U A ' B '
from which (1) is immediate upon taking probabilities of both sides of
(3) and noting that AB and A ' B ' are disjoint or mutually exclusive
and that the probability of disjoint set unions is additive.
The set (X<-->Y) is just defined as (A<-->B) for A = {X < = x}, B = {Y
< = y}, AB = {X < = x, Y < = y}. A ' B ' = {X > x, Y > y}, and the
notation {X < = x} means {w: X(w) < = x} for example, where w is any
point in the probability space satisfying the inequality.
From (2) we see that Probable Correlation of X and Y, defined as P(X<--
Y) for X, Y continuous random variables, is symmetric in the
arguments x, y. The second partial derivative of P(X<-->Y) from (2)
is:
4) Dxy[P(X<-->Y)] = f(x, y) + Dxy[P(X > x, Y > y)]
So we have here a generalization of the mixed partial derivative
equation for P(X-->Y). The part of the solution that yields f(x, y)
is the same as before, and now we have the second term on the right
hand side of (4) to contend with.
We'll leave this topic here for now, except to note a few things
including to note that:
5) (AB) U (A ' B ' ) U (AB ' ) U (A ' B) = universe
and since all sets in (5) on the left hand side are disjoint, we get:
6) P(AB) + P(A ' B ' ) + P(AB ' ) + P(A ' B) = 1
and the first two probabilities of (6) in terms of probabilities
related to random variables X and Y are F(x, y) and P(X > x, Y > y).
The third is P(X < = x, Y > y), the fourth P(X > x, Y < = y).
Notice that:
7) P(X > x) = P(X > x, Y > y) + P(X > x, Y < = y)
and therefore:
8) 1 - FX(x) = P(X > x, Y > y) + P(X > x, Y < = y)
Also:
9) P(Y > y) = P(X > x, Y > y) + P(X < = x, Y > y)
and therefore:
10) 1 - FY(y) = P(X > x, Y > y) + P(X < = x, Y > y)
where FX(x) = P(X < = x) and FY(y) = P(Y < = y).
Osher Doctorow
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