(the maths inside is not meant to be taken seriously;
'tis a silly idea that can't work)
from a recent email by Peter Broadbery
Random variables are assumed to have the following properties:
- complex constants are random variables;
- the sum of two random variables is a random variable;
- the product of two random variables is a random variable;
- addition and multiplication of random variables are both commutative; and
- there is a notion of conjugation of random variables, satisfying:
and 
for all random variables 
, 
, and coinciding with complex conjugation
if is a constant.
This means that random variables form complex abelian 
-algebras. If 
,
the random variable a is called "real".
An expectation E on an algebra A of random variables is a normalized, positive
linear functional. What this means is that
-
;
-
for all random variables ;
-
for all random variables and ; and
-
if 
is a constant.
-algebra
From Wikipedia, the free encyclopedia
In mathematics, a -algebra is an associative algebra over the field of
complex numbers with an antilinear, antiautomorphism 
which is an
involution. More precisely, is required to satisfy the following properties:
for all 
,
in 
, and all in 
.
The most obvious example of a -algebra is the field of complex numbers C
where is just complex conjugation. Another example is the algebra of nn
matrices over with given by the conjugate transpose.
An algebra homomorphism 
is a -homomorphism if it is compatible
with the involutions of and 
, i.e.
An element in is called self-adjoint if 
.
aldor
#include "axiom"
RandomAlgebra(F: Field): Category == with {
Algebra F;
E: % -> F;
sample: % -> F;
}
local PolyHelper(F: Field): with {
expand: SparseUnivariatePolynomial F -> Generator Cross(F, NonNegativeInteger);
}
== add {
expand(p: SparseUnivariatePolynomial F): Generator Cross(F, NonNegativeInteger) == generate {
default m: SparseUnivariatePolynomial F;
import from SparseUnivariatePolynomial F;
import from List SparseUnivariatePolynomial F;
for m in monomials p repeat {
yield (leadingCoefficient m, degree m);
}
}
}
UnivariateNormalRandomAlgebra: RandomAlgebra Float with {
X: () -> %;
variance: % -> Float;
} == add {
Rep ==> SparseUnivariatePolynomial Float;
import from Rep;
0: % == per 0;
1: % == per 1;
X(): % == per(monomial(1$Float,1$NonNegativeInteger)$Rep);
characteristic(): NonNegativeInteger == 0;
-(x: %): % == per(-rep x);
(a: %) = (b: %): Boolean == rep(a) = rep(b);
(a: %) + (b: %): % == per(rep(a) + rep(b));
(a: %) * (b: %): % == per(rep(a) * rep(b));
(a: Float) * (b: %): % == per(a * rep(b));
coerce(x: Integer): % == per(x::Rep);
coerce(x: Float): % == per(x::Rep);
coerce(x: %): OutputForm == coerce rep(x);
E(X: %): Float == {
import from PolyHelper Float;
z: Float := 0;
for p in expand rep(X) repeat {
(a, b) := p;
z := z + a * E(b);
}
z
}
-- should be a random sampling of x.
sample(X: %): Float == {
import from PolyHelper Float;
import from Float;
u := uniform01()$RandomFloatDistributions;
x: Float := 0;
for p in expand rep(X) repeat {
(a, b) := p;
x := x + a * u^b;
}
return x;
}
variance(X: %): Float == { A := (X-E(X)*1); E(A*A); }
-- return expected value of X^n
local E(n: NonNegativeInteger): Float == {
p: Rep := 1;
-- yuck. There must be a nicer way than this..
for i in 1..n repeat p := differentiate(p) + monomial(1,1)*p;
coefficient(p,0);
}
}
aldor
Compiling FriCAS source code from file
/var/zope2/var/LatexWiki/2618023199496535153-25px001.as using
AXIOM-XL compiler and options
-O -Fasy -Fao -Flsp -laxiom -Mno-AXL_W_WillObsolete -DAxiom -Y $AXIOM/algebra
Use the system command )set compiler args to change these
options.
#1 (Warning) Deprecated message prefix: use `ALDOR_' instead of `_AXL'
Compiling Lisp source code from file
./2618023199496535153-25px001.lsp
Issuing )library command for 2618023199496535153-25px001
Reading /var/zope2/var/LatexWiki/2618023199496535153-25px001.asy
UnivariateNormalRandomAlgebra is now explicitly exposed in frame
initial
UnivariateNormalRandomAlgebra will be automatically loaded when
needed from /var/zope2/var/LatexWiki/2618023199496535153-25px001
RandomAlgebra is now explicitly exposed in frame initial
RandomAlgebra will be automatically loaded when needed from
/var/zope2/var/LatexWiki/2618023199496535153-25px001axiom
a := X()$UnivariateNormalRandomAlgebra
Type: UnivariateNormalRandomAlgebra
?axiom
-- a number, normally distributed
sample a
Type: Float
Type: Float
Type: Float
Type: Float
Type: Float
axiom
-- 3, apparently
variance(a^2 + a)
Type: Float
for all 
