Re: Independent/Dependent Phases 9: Mixed Partial Physics Equations

>>From Osher Doctorow mdoctorow@xxxxxxxxxxx

Since the three probability expressions apply to physics:

1) P(X-->Y)(x,y) = 1 + F(x,y) - FX(x)
2) FY|X(y|x) = F(x,y)/FX(x) if FX(x) is not 0
3) FY|X(y|x) = FY(y) if X, Y are statistically independent

let's take a look at the corresponding Dxy expressions.

4) Dxy[P(X-->Y)(x,y) = f(x,y)

This is because Dxy[F(x,y)] = f(x,y) from well-known mathematical
probability-statistics for continuous random variables X, Y.

So we're not quite depending on particular examples to claim that Dxy
is important, since f(x,y) is the bivariate probability density
function of X and Y and is of key importance in mathematical
probability-statistics and its applications to physics.

Let's calculate Dxy[FY|X(y|x)].

1) Dx[F(x,y)/FX(x)] = [FX(x)Dx(F(x,y)) - F(x,y)fX(x)]/FX(x)^2

2) Dy[Dx[F(x,y)/FX(x)] = [FX(x)f(x,y) - fX(x)DyF(x,y)]/FX(x)^2

The second expression reduces to f(x,y)/FX(x) - fX(x)DyF(x,y)/FX(x)^2
and is unlikely to have any probability-statistics interpretation.of
simple type.

Finally, for Dxy[FY(y)] we get:

3) Dxy[FY(y)] = 0

which is simple.

Thus, for Independent Probability-Statistics and for Probable Influence
(PI), we get simple expressions 0 and f(x,y) respectively in applying
Dxy to the relevant quantities FY|X(y/x) (often written FY|X=x (y|x))
and P(X-->Y)(x,y), but for conditional probability FY|X=x(y|x) when Y
and X are not statistically independent, we get the complicated
expression which is not only arguably uninterpretable physically but
does not have simplicity.

The simplicity and meaningfulness of interpretation of Dyt(y) = 1/2 =
Dtt(y) for the Riccati Differential equation for the uniform and
exponential distributions compared to the normal/Gaussian distribution
(it fails for the latter, as Puppet_Sock noticed) becomes more
interesting now. Readers may recall that the uniform and exponential
distributions among others are ranked higher in PI maximum entropy than
the normal/Gaussian distribution. In fact, as numerous previous posts
of mine have described, the uniform and other finite interval
distributions are highest in PI maximum entropy ranking, the
nonnegative line distributions such as the exponential and more general
gamma and F distributions are second, and the symmetric distributions
like the normal/Gaussian are last. The ranking is the same as that by
Dxy in simplicity and physical meaningfulness.

Osher Doctorow

.

Relevant Pages