(the maths inside is not meant to be taken seriously; 'tis a silly idea that can't work) Note that dropping comutativity below we get quantum probablility. Franz Lehner added packages supporting computations for several popular versions. In principle one can treat commutative case as a special case of quantum probablility...
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 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. - for all in .
An element in is called self-adjoint if . aldor #include "axiom" aldor Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2618023199496535153-25px001.as using Aldor compiler and options -O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra Use the system command )set compiler args to change these options. The )library system command was not called after compilation. fricas a := X()$UnivariateNormalRandomAlgebra |