Acceleration of Universe As Acceleration of Probable Correlation 2: An Example
- From: "OsherD" <mdoctorow@xxxxxxxxxxx>
- Date: 22 Sep 2005 00:22:26 -0700
>>From Osher Doctorow mdoctorow@xxxxxxxxxxx
Let's consider one of my favorite examples of a bivariate joint cdf
F(x,y) (cumulative distribution function) and pdf f(x,y) (probability
density function), GUmbel's bivariate exponential:
1) F(x,y) = 1 - exp(-x) - exp(-y) + exp(-(x + y + kxy))
where k (also written theta) is in [0, 1] and x, y are positive. Its
univariate marginals are each standard exponential, and its pdf is:
2) f(x,y) = exp(-x - y - kxy){(1 + ky)(1 + ky) - k}
If k is 0, X and Y are statistically independent.
My reason for preferring this distribution to most other bivariate
distributions at least for theory is that it's simple compared to most
bivariate joint cdfs and pdfs. It lacks some advantages such as
"comprehensivenss" (look it up on the internet as a keyword, or in one
of my earlier postings).
Now let's look at Probable Correlation:
3) P(X<-->Y)(x,y) = F(x,y) + R(x,y)
where bivariate joint reliability (in engineering) R(x,y) is P(X > x, Y
> y) for X, Y continuous random variables as here.
We can substitute F(x,y) from (1) above for this example, but what
should we take for R(x,y)? One fascinating way of getting examples of
R(x,y) is to just postulate or assume symmetry, that is to say:
4) R(x,y) = F(x,y)
which says:
5) P(X > x, Y > y) = P(X < = x, Y < = y)
Because of (4) and (5), (3) becomes for this assumption:
60 P(X<-->Y)(x,y) = 2F(x,y)
and therefore the second mixed partial derivative Dxy of P(X<-->Y)(x,y)
is just 2f(x,y) where f(x,y) is nonnegative from (2). Since
Dxy[P(X<-->Y)] is taken to be the analog of the physical acceleration
of the universe, it's easy to see that a wide variety of scenarios have
f(x,y) > 0 on an interval from (2) and therefore Dxy of P(X<-->Y) is
greater than 0 there.
In addition, we see how the probability distribution could change at
the commencement of each acceleration interval (from a lower
distribution on the previous interval), for example by k undergoing a
change from a value k1 to a different value k2. It's easy to see or
prove by differentiation that F(x,y) decreases as k increases from 0 to
1, so if k becomes a little smaller, F(x,y) increases on the interval
if it isn't too close to 1 (F(x,y) being bounded above by least upper
bound 1). A change from bigger to smaller k would presumably be
abrupt, but nothing prevents probability distributions from undergoing
abrupt changes in time. It is even hypothetically possible for k to be
some "almost constant" function of time that just changes at a few
discrete times.
Osher Doctorow
.
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