Natural Science Forum / Physics / General Physics / July 2008

From Osher Doctorow

Let's examine the relationship between Probable Correlation in
Probable Causation/Influence (PI) and the Linear Population
Correlation Coefficient (also called Linear Correlation) where the
latter is in Mainstream Probability-Statistics.   The Latter is
defined as rho, given by:

1) rho = [E[(X - E(X))(Y - E(Y))]/product of standard deviations of X
and Y

The former is:

2) P(A<-->B) = P(AB) + P(A ' B ' )

Let's assume that X and Y and standardized or normalized (in units of
standard deviations, in other words in this type of case) in (1) and
without loss of generality that their respective population means E(X)
= 0 and E(Y) = 0, thus reducing rho of (1) to:

3) rho = E(XY) for X, Y standardized with means 0

The nearest pointwise expression of which E(XY) is the average or mean
is:

4) kernel(rho) = XY

Now, in the usual notation, the Cumulative Distribution Function (cdf)
of continuous random variable X, FX(x), at point x, is:

5) FX(x) = P(X < = x)

and likewise for Y:

6) FY(y) = P(Y < = y)

and the joint cdf F(x, y) of X and Y is:

7) F(x, y) = P(X < = x, Y < = y)

To see how A and B of (2) are related to X and Y, the usual PI
procedure is to define:

8) A = {w: X(w) < = x}
9) B = {w: Y(w) < = y}

so that:

10) FX(x) = P(A), FY(y) = P(B), F(x, y) = P(AB)

Therefore, from (2):

11) P(A<-->B) = P(AB) + P(A ' B ' ) = F(x, y) + P(A ' B ' )

It turns out that as X increases, that is to say the value of X(w) = x
increases, then F(x, y) for y fixed increases or remains constant, and
so does FX(x).  Similarly as Y increases for F(x, y) for x fixed as
well as FY(y).   But the quantity XY of (4) has the same property for
X and Y both nonnegative or both nonpositive.  Therefore:

12) both P(A<-->B) (which in this type of situation is written as
P(X<-->Y) and rho or rho_XY, the respective Probable Correlation of X
and Y and the Linear Correlation (Coefficient) of X and Y, increase
together under the general conditions of the (3) statements below (11)
above.

This can also be rephrased in terms of partial derivatives being
nonnegative.

Osher Doctorow