Talk:Algebra of random variables

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Needs radical repair

As of January 2013, this article is in urgent need of a complete rewrite. The article mixes exposition at several different levels of sophistication, and makes several serious blunders. This degree of confusion is unacceptable in a mathematical Wikipedia article. —Aetheling (talk) 07:16, 5 January 2013 (UTC)[reply]

January 2013 according to the signature. =P Anyway, I agree and I've tagged it appropriately to hopefully increase its visibility in relevant cleanup categories and the Statistics WikiProject. To document my own reasoning with regard to the tags: the article provides very little general context and discussion accessible to someone who's unfamiliar with it, nor a solid or well-organized overview, nor even any kind of summary in the lead paragraph. It dives into various details and, as the above user said, mixes information at different levels of mathematical sophistication to the point that it's utterly impossible to follow without having a solid background already. Laogeodritt ^{[ Talk | Contribs ]} 17:25, 3 February 2013 (UTC)[reply]

von Neumann algebra

Shouldn't the *-algebra be a Von Neumann algebra to take into account infinite dimensional spaces?

Conditional expectation

Is there a similar axiomatization for conditional expectation? -- Spireguy (talk) 02:36, 26 March 2010 (UTC)[reply]

One definition of the quantum (non-commutative) version of a conditional expectation is

"Let there be a quantum probability space (N,ρ) and a commutative von Neumann sub-algebra C ⊂ N. Then the quantum conditional expectation π( · ) is a map from C′ (the commutant of C) to C such that ρ (π(X)Z) = ρ (XZ) for all X ∈ C′, Z ∈ C."

So I'd hazard a guess that an appropriate algebraic definition for the (commutative) conditional expectation would be

"If N is the algebra of all random variables considered, and C is a (von Neumann) sub-algebra (relating to the measurement outcome), then the conditional expectation is a map P from N to C such that E[P(X)Z] = E[XZ] for all X in N and Z in C"

Thoughts? -- S.Wilson (talk) 10:46, 9 May 2011 (UTC)[reply]

Rules for variance

Could someone (who knows for sure) add the rules for variance on random variables. E.g.:

Var(c*X) = c*Var(x)

213.165.179.229 (talk) 18:50, 10 July 2011 (UTC)[reply]

That is incorrect. Should be c^2*Var[X], but that is covered in the article on variance under the basic properties section.50.147.26.108 (talk) 05:14, 10 January 2014 (UTC)[reply]

Applicability to nonnumeric outcomes

What random variable algebra would cater to a fair coin, or two coins, or a 4×4 grid of boggle dice, when there is no numeric outcome? Does one arbitrarily assign each of the n outcomes to n distinct complex numbers (e.g. 1 for heads and 0 for tails, 1-26 for A-Z), or to the unit vectors of an n-dimensional Hilbert space as in quantum mechanics, or does one count the number of heads and tails in a sequence, or all of the above, or is the whole theory simply not applicable to nonnumeric outcomes, or what?

Also, why the requirement that every complex number be a variable? Why can't one have a real random variable algebra?

Why no requirement of associativity for addition and multiplication, or that multiplication distribute over addition, or that (a+b)* = a*+b*, or that a*a (product of a* and a) be real (without which 2 makes no sense)?

What statistical notions are definable in a random variable algebra? Is there sufficient structure to define linearly (un)correlated, Pearson correlation coefficient, statistically independent, distance correlation, etc.?

What is the simplest nontrivial algebra of random variables? 2D complex Hilbert space? Made an algebra how (what product)? Or if there are real random variable algebras then 2D real Hilbert space? Vaughan Pratt (talk) 23:04, 18 July 2014 (UTC)[reply]