Quantum Gravity 281.91: Correction on Kotz and Johnson

From Osher Doctorow

What Kotz and Johnson proved (around the 1980s or early 1990s to my
recollection - they are the authors of the famous Wiley series on
bivariate and univariate distributions) is actually:

1) max(FX(x) + FY(y) - 1, 0) < = F(x, y) < = min(FX(x), FY(y))

rather than the same thing with fX, fY, f(x, y) respectively replacing
FX, FY, F(x, y).

Therefore, the theorems in 281.90 and 281.9 pertaining to the latter
have to make the assumption that:

2) fX(x) + fY(y) - 1 < = f(x, y) (assumption)

rather than considering it verified. This is plausible for a wide
variety of scenarios and probability density functions (pdfs), but it
presumably fails with certain densities under certain conditions to be
specified. Also, as with FX(x) + FY(y) - 1, the left hand side of (2)
may be either positive or negative or 0 depending on the pdf.

Notice that (2) is equivalent to:

3) fX(x) + fY(y) - f(x, y) < = 1 (assumption)

which is reminiscent of:

4) P(A U B) = P(A) + P(B) - P(AB) < = 1

and in fact when A = {w: X(w) < = x}, B = {w: Y(w) < = y}, then AB =
{w: X < = x, Y < = y}, so P(A) = FX(x), P(B) = FY(y), P(AB) = F(x, y),
which proves as in Kotz and Johnson that:

5) FX(x) + FY(y) - F(x, y) < = 1

which is part of (1). The basic reason why fX may differ from FX and
so on for the other quantities is roughly that, unlike FX, fX
sometimes for certain pdfs and scenario exceeds 1 when the pdf is
sharply peaked and narrow (small variance) around its mean or median.

Osher Doctorow
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