Talk:Algebra of random variables

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)

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)

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)

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)

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)

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)

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)

Very interesting publishing enterprise

While trying to find anything at all about this subject, I followed up the three references in the article. The only promising one was the book of that title (plus "The") by Melvin Dale Springer, 1918-2008. Amazon offers 6 used copies from various vendors at prices from $199.00 to $3,115.00 (from a vendor with many books at almost exactly that price).

So then I went to eBay to see if anyone had a better price on it. No, but I did find a 2012 paperback of the same name as the Wikipedia article by Jesse Russell and Ronald Cohn for only $28.99. Since that was way cheaper I decided to buy it, but checked the page count first. 82 pages. Well, a bit short.

But then I noticed the product description: "High Quality Content by WIKIPEDIA articles! In the algebraic axiomatization of probability theory, the primary concept is not that of probability of an event, but rather that of a random variable. Probability distributions are determined by assigning an expectation to each random variable. The measurable space and the probability measure arise from the random variables and expectations by means of well-known representation theorems of analysis. One of the important features of the algebraic approach is that apparently infinite-dimensional probability distributions are not harder to formalize than finite-dimensional ones. This book was created using print-on-demand technology."

That sure rang a bell. But perhaps Russell and Cohn are such great math expositors that 82 pages by them are well worth the investment. Googling them almost immediately brought me to this page.

Apparently Russell and Cohn have created hundreds (thousands?) of books by collecting Wikipedia articles that someone (a computer program?) has judged to be related enough to be bound together as one book under the title of one of them.

Abe Books has two examples at the top of their list. Since the first book there, Ulmus davidiana var. jopanica, is published by Book on Demand, strictly speaking it isn't even necessary to spend time deciding which articles to include until someone actually orders a copy since no one knows what's in most of these books until one is ordered, not even their "authors". After selling one, subsequent copies can then use the same selection, both for consistency and to save the work of selecting again.

A catalog could include titles of every single Wikipedia article, all with around 80 pages, which would be made up by automatically selecting enough related articles when ordered.

Seems like a sound business plan. Anyone see anything wrong with it? ;)

In the end I decided not to buy the book. Vaughan Pratt (talk) 00:57, 19 July 2014 (UTC)

The Creative Commons license used by Wikipedia explicitly allows this practice, and in a similar vein to the "Mozilla Firefox" CDs sold at some computer stores, jackasses will be jackasses. Thanks for the heads up though, and consider adding the publisher name to the list in the aforementioned article and the book name to the list on PrimeHunter's page. SamuelRiv (talk) 04:37, 20 July 2014 (UTC)

Approximations by Taylor series expansions of moments

Most statements in this section are either wrong, or meaningless (see Moment problem#Uniqueness_(or_determinacy)). It should be corrected marking which parts are valid in which contexts, and which have value only as heuristics. --Ilya-zz (talk) 06:11, 16 March 2021 (UTC)