Re: Dynamical Systems and Expansion-Contraction

>>From Osher Doctorow

Sometimes for two random variables X1 and X2, we have FX1(x) > FX2(x)
for a certain sub-interval of x values but the opposite for another
sub-interval, which is itself quite interesting.

For example, let X1 and X2 be respectively the standard uniform and
standard exponential random variables, respectively on (0, 1) or [0, 1]
versus on [0, infinity) with the latter having parameter 1:

1) FX1(x) = x
2) FX2(x) = 1 - exp(-x)

We can then prove that on [0.65, 1.0] we have:

3) x > 1 - exp(-x) on [0.65, 1.0]

so that FX1(x) > FX2(x) there, but the situation is quite different on
[0, 0.65).

If x = 0.65, then exp(-x) is .5220 approximately so that x + exp(-x) >
1 as required by (3). The derivative of x + exp(-x) is:

4) d(x + exp(-x))/dx = 1 - exp(-x) > 0

since exp(-x) = 1/exp(x) < 1 iff exp(x) > 1 iff x > 0.

Therefore, at x = 0.65, we have x + exp(--x) > 1, and by (4) x +
exp(-x) > 1 for x > 0.65.

The point of this is that even for univariate distributions, "analogs"
of inequalities such as those required to establish that Probable
Correlation P(X<-->Y) = F(x, y) + P(X > x, Y > y) exceeds the Probable
Correlation for other pairs of variables such as (Y, Z), (X, Z), (U,
V), etc., can be found even if for a smaller interval than the whole
range or domain.

The pointwise definition of Probable Correlation P(X<-->Y)(x, y)
(abbreviated P(X<-->Y) if the values are understood) thus can enable
focusing on more interesting or important regions where correlation is
higher in certain cases.

Osher Doctorow

.

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