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(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:

  1. complex constants are random variables;
  2. the sum of two random variables is a random variable;
  3. the product of two random variables is a random variable;
  4. addition and multiplication of random variables are both commutative; and
  5. there is a notion of conjugation of random variables, satisfying:

    (ab)<em> = b</em> a<em> and a** = a

    for all random variables a, b, and coinciding with complex conjugation if a is a constant.

This means that random variables form complex abelian *-algebras. If a = a*, 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

  1. E(1) = 1;
  2. E(a <em> a) >= 0 for all random variables a;
  3. E(a + b) = E(a) + E(b) for all random variables a and b; and
  4. E(za) = zE(a) if z is a constant.


From Wikipedia, the free encyclopedia

In mathematics, a *-algebra is an associative algebra over the field of complex numbers with an antilinear, antiautomorphism <em> : A \rightarrow A which is an involution. More precisely, * is required to satisfy the following properties:

  • (x + y)^<em> = x^</em> + y^<em> \quad
  • (z x)^<em> = \overline{z} x^</em>
  • (x y)^<em> = y^</em> x^<em> \quad
  • (x^<em>)^</em> = x \quad

for all x,y in A, and all z in C.

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 C with * given by the conjugate transpose.

An algebra homomorphism f : A \rightarrow B is a *-homomorphism if it is compatible with the involutions of A and B, i.e.

  • f(a <em> ) = f(a) </em> for all a in A.

An element a in A is called self-adjoint if a* = a.

#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); } }
   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-ALDOR_W_WillObsolete -DAxiom -Y $AXIOM/algebra -I $AXIOM/algebra
      Use the system command )set compiler args to change these 
   Compiling Lisp source code from file 
   Issuing )library command for 2618023199496535153-25px001
   Reading /var/zope2/var/LatexWiki/2618023199496535153-25px001.asy
   RandomAlgebra is now explicitly exposed in frame initial 
   RandomAlgebra will be automatically loaded when needed from 
   UnivariateNormalRandomAlgebra is now explicitly exposed in frame 
   UnivariateNormalRandomAlgebra will be automatically loaded when 
      needed from /var/zope2/var/LatexWiki/2618023199496535153-25px001
>> System error: The bounding indices 163 and 162 are bad for a sequence of length 162. See also: The ANSI Standard, Glossary entry for "bounding index designator" The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERROR

a := X()$UnivariateNormalRandomAlgebra

Type: UnivariateNormalRandomAlgebra?
-- a number, normally distributed 
sample a

Type: Float
-- 0
E a

Type: Float
-- 1

Type: Float
-- 1

Type: Float
-- 5

Type: Float
-- 3, apparently
variance(a^2 + a)

Type: Float

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