| |
 |
|
|
Science Forum Index » Physics Forum » Quantum Gravity 281.93: F(x, y) < = (x o y)_F < = xy +...
Page 1 of 1
|
| Author |
Message |
| OsherD... |
 Posted: Sun Jul 20, 2008 5:54 pm |
|
|
|
Guest
|
From Osher Doctorow
Recall that (with X, Y continuous random variables here):
1) (x o y)_F (or (x oF y) ) = (definition) xy + F(x, y) - yFX(x)
To prove that:
2) (x o y)_F < = xy + F(x, y),
notice that:
3) (x o y)_F = F(x, y) + y(x - FX(x)) < = F(x, y) + xy
since 0 < = x < = 1 and 0 < = FX(x) < = 1 by definition.
Next, to prove that:
4) F(x, y) < = (x o y)_F = xy + F(x, y) - yFX(x) if fX(u) < = 1 for
all u
we have from the mean value theorem for integrals on [0, x]:
5) FX(x) = I fX(u)du = xfX(z) < = x where I...du is the integral on
[0, x] and 0 < z < x provided that fX(u) < = 1 for all u (which is
very common)
and therefore from (5):
6) -x < = -FX(x), so -xy < = -yFX(x)
and therefore from the definition of (x o y)_F:
7) (x o y)_F = xy + F(x, y) - yFX(x) > = xy + F(x, y) - xy = F(x, y)
if fX(u) < = 1 for all u
This proves (4). We therefore have proven:
8) F(x, y) < = (x o y)_F < = xy + F(x, y) if fX(u) < = 1 for all u, X
and Y continuous random variables.
Osher Doctorow |
|
|
| Back to top |
|
| |
|
Page 1 of 1
All times are GMT - 5 Hours
The time now is Mon Oct 13, 2008 12:49 am
|
|