Re: Dynamical Systems and Expansion-Contraction

>>From Osher Doctorow

Let's take a very simple example.

The uniform marginal cdf on the interval [a, b] or (a, b) is:

1) FX(x) = (x - a)/(b - a)

Depending on a and b (constants or "parameters"), a different member of
the uniform family is invovled. Let's call FX1 the uniform marginal on
[0, 1] so that a = 0, b = 1, and let's call FX2 the uniform marginal on
[0, b] or (0, b) for arbitrary positive b. Let's examine when we
have:

2) FX1(x) > = FX2(x)

which says:

3) x > x/b

since (x - 0)/(1 - 0) = x and (x - 0)/(b - 0) is x/b.

If we restrict x to (0, 1), then dividing both sides of (3) by x
yields:

4) 1 > 1/b

and multiplying both sides by b > 0 yields:

5) b > 1

Therefore, for b > 1, FX1 on (0, 1) > = FX2 on (0, b).

The same idea can be used for bivariate cdfsF(x, y) and for pdfs fX(x),
f(x, y), etc.

Osher Doctorow

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