Science > Physics > Independent/Dependent Phases 4: Some f(x) = 1/2 Comparisons
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Science > Physics |
| User: |
"OsherD" |
| Date: |
04 Oct 2005 09:10:42 PM |
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Independent/Dependent Phases 4: Some f(x) = 1/2 Comparisons |
From Osher Doctorow
A. The uniform pdf on interval [a, b]:
1) f(x) or fX(x) = 1/(b - a) on [a, b], = 0 elsewhere
So for fX(x) = 1/2 we get:
2) 1/(b - a) = 1/2
which is equivalent to:
3) b - a = 2
If we put a = 0 for convenience, we get for that special case:
4) b = 2
So fX(x) = 1/2 occurs for the uniform distribution on [0, 2] or on [a,
a + 2] for any real a. Interestingly, it does not occur for any
uniform distribution on [a, b] with b - a < 2, as for example on [0, 1]
or [0, 1/2] or [0, 1/10], etc.
If X were a random time with values from 0 upward, or in stochastic
notation {X_t} would have respective fX_t(x) pdfs, then if X_0 had
[a_0, b_0] such that b_0 - a_0 << 2, then if intervals widen as t
increases, for some t we would reach b_t - a_t = 2 and then b_t' - a_t'
would increase for t' > t.
Does this mean anything physically?
Well, I think so. Remember that independent events A and B are such
that B is not "affected" at all by A having occurred previously. So
let's call this 0-step influence. Markov chains are such that event B
is only influenced by "one-step-previous" events A, as for example the
next checkers or chess move being determined only by the present state
or move on the board. So call this 1-step influence. Markov chains
are the "minimal dependent" chains in this sense. Chains with "more
than 1-step memory" or for short "memory" are influenced by 2 or more
past events. So 2 is the boundary of Non-Markov Non-Independent
events/processes/chains in terms of "number of past steps".
In this sense, we can say that if the Universe changes its acceleration
from negative to positive or vice versa at fX(x) = 1/2 where X is a
time random variable, then the Universe arguably can have changed from
"Non-Memory" to "Memory" with these qualifications (or vice-versa).
Osher Doctorow
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| User: "" |
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| Title: Re: Independent/Dependent Phases 4: Some f(x) = 1/2 Comparisons |
04 Oct 2005 09:14:46 PM |
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You may be right.
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| User: "OsherD" |
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| Title: Re: Independent/Dependent Phases 4: Some f(x) = 1/2 Comparisons |
04 Oct 2005 09:33:56 PM |
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From Osher Doctorow
dostockba typed:
You may be right.
Thanks. It does seem to be an interesting idea. In many of my Rare
Event threads starting with geometry.research, sci.stat.math,
math-history-list at Math Forum, real_analysis (I forget its full name,
but it's at U. Louisville, Kentucky), etc., I used to define Rare
Events/Processes as involving influence by 2 or more previous
Events/objects/etc. There's arguably something to that idea as well as
this one on uniform distributions.
On the other hand, let's look at:
B. Standard exponential pdf:
1) fX(x) = exp(-x)
For fX(x) = 1/2, we get:
2) 1/exp(x) = 1/2
and therefore:
3) exp(x) = 2
and so:
4) x = log(2)
where natural logs are used here. This is up to a constant the natural
unit of Shannon information/entropy or can be taken as the "kernel" of
its operator. In fact, Shannon information/entropy's predecessor,
Hartley information/entropy, is just log_2(p) where log_2 is logarithm
to base 2 and p is probability.
Aside from this, however, recall that:
5) log(1) = 0
and therefore log(2) to the same base e is at least in integer units
the next "highest" logarithm. But 2 isn't a probability. However,
neither is [a, b]. So we have some fascinating and possibly related
directions of inquiry.
Osher Doctorow
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