Quantum Gravity 295.3: Maximizing q(x, y) at F = 1/2 or p = 1/2
- From: OsherD <mdoctorow@xxxxxxxxx>
- Date: Thu, 30 Oct 2008 23:25:32 -0700 (PDT)
From Osher Doctorow
Regarding:
1) q(x, y) = ap(1 - p) + cF(1 - F), a and c constants
from 295.2, we can easily see that for p and F regarded as independent
variables, q is locally maximized at p = 1/2 for F fixed and F = 1/2
for p fixed:
2) q(x, y) has a local maximum with respect to p at p = 1/2 for F
fixed and with respect to F at F = 1/2 for p fixed.
by the second derivative test for maxima and minima. The value of
q(x, y) if both expressions are maximum (at p = 1/2 and F = 1/2) is:
3) q(x, y) = (1/4)(a + c) at p = 1/2 and F = 1/2
If a and c are both equal, that is to say p and F contribute equally
to q, then:
4) q(x, y) = (1/4)a at p = 1/2 and F = 1/2 for a = c
For example, if a = 1/4, then q(x, y) = 1/16 gives the local maximum
at p = 1/2 and F = 1/2 for a = c. Since we don't arguably consider a
or c to be greater than 1, the local maximum of q is more on the 0
side than the 1 side of the interval [0, 1].
If we regard q(x, y) as a function of p and F, written q(p, F) for
simplicity, then q(1/2, 1/2) is the value of q(x, y) at the median of
the joint distribution of X and Y (the median is defined by F = 1/2,
or technically (x, y) or here (p, q) such that F = 1/2) with the
additional condition that p = 1/2 to locally maximize q(x, y) or q(p,
F).
Osher Doctorow
.
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