Talk:Probability theory

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Mathematics grading: B Class High Importance  Field: Probability and statistics

Needs general expansion needed, plus more on history and applications. Tompw 16:33, 7 October 2006 (UTC)

This page appears to be mostly or entirely redundant with the probability page. I'd suggest that we merge any differences into probability and remove this page. Comments? Wile E. Heresiarch 15:14, 27 Dec 2003 (UTC)

OK, on second thought it does make sense to have a separate page for probability theory. Other pages can refer specifically theoretical issues to the theory page. 128.138.85.48 02:25, 3 Jan 2004 (UTC)

The external link is broken.

Probability Theory shouldn't be included in the Discrete Math Category. Dennis 17:40, 16 Dec 2004 (UTC)

I disagree. First, probability in discrete spaces has peculiar features distinct from continuous spaces. In addition, many of the techniques of finite mathematics such as difference equations, generating functions, and the like, originated in probability theory (a classic survey is Laplace's Analytical Theory of Probability, written in 1812!!) and probability techniques are sometimes important in numbers theory. Finally, using his own version of nonstandard analysis, Edward Nelson reduced all of probability theory (including stochastic processes) to discrete probability spaces(this is in his book Radically Elementary Probability Theory). — Miguel 04:52, 2004 Dec 19 (UTC)

Why is this page so biased towards Bayesian statistics? INic 12:08, 19 October 2005 (UTC)

Contents

[edit] sequences?

in the explanation of sample space, shouldn't the word "sequence" be "combination" as the order of the Californian voters does not matter?

[edit] important article

In response to the comment about whether this page is the same as the article on probability: this page is certainly not the same as the other article. Kolmogorov's axioms are stated clearly here, and it is important that they are in the Wikipedia. This article is quite good and very important, but it still would benefit from some editing. For example, Ω is the "probability space", while "sample space" S comes into play once a random variable X (or Y or Z or whatever you'd like to name it), which is a measurable function, maps X:Ω→S where S is the space from which we collect samples. What is the difference, one might ask, but truly, this is important mathematically. Ω could be anything. As examples, Ω could be {the people of earth}, {the atoms in the cosmos}, {all 18K gold jewelry or golden crowns}, or any such set of interest. But the sample space S would be something such as, respectively, the set of vectors (height, weight, age) of the people; the set of number of electrons of the atoms and their quantum energy levels; the set of all weights of the golden items. There are some other improvements that are warranted, too. I plan to begin editing the article soon, but will wait for further discussion, so that we can do the best job of doing so as a team. -- MathStatWoman

With the most recent edits, I wonder whether this page continnues to serve a useful pupose. Rjm at sleepers 09:24, 10 March 2007 (UTC)

[edit] Don't use F to mean two completely different things

I'm going to remove all references in this page to F as an event. It will only confuse people later when they see that F is a sigma-algebra. E=>A and F=>B is my suggested universal fix.

[edit] Probability is related to life

The article on probability theory is superficial. It uses jargon, while being disconnected from real life. I believe that the best foundation to theory of probability is laid out here:

[1]

The article is accompanied by free software pertinent to probability (combinatorics and statistics as well).

Ion Saliu, Probably At-Large

[edit] Proposed with probability axioms

This page currently contains two main sections:

I propose that this page is merged with probability axioms and limits its discussion to the first bullet point. Perhaps it should be renamed 'Kolmogorov axioms' (a title which currently redirects to probability axioms)? Andeggs 16:18, 24 December 2006 (UTC)

I agree; there is a great deal of redundancy in having two different articles on the Kologrov axioms; this article should be merged with probability axioms and probability interpretations.

-Patrick N.R. Julius 03:00, 28 December 2006 (UTC)

"Probability theory" should redirect to probability if the proposed move of this article's material to those other two articles gets done. Michael Hardy 00:13, 29 December 2006 (UTC)
You're right there's tonnes of redundancy. I'm going to merge all of the information about the probability axioms along with probability space into this page, and then we can decide where that should go. I hope that I'm not stepping on anyone's toes :) MisterSheik 14:11, 28 February 2007 (UTC)
Done! MisterSheik 17:59, 28 February 2007 (UTC)
I moved this from the introduction because it wasn't clear to me what it actually meant. The definition in the article seems clear and rigorous by comparison. MisterSheik 18:18, 28 February 2007 (UTC)

More precisely, probability is used for modelling situations when the result of an experiment, realized under the same circumstances, produces different results (typically throwing a dice or a coin). Mathematicians and actuaries think of probabilities as numbers in the closed interval from 0 to 1 assigned to "events" whose occurrence or failure to occur is random. Probabilities P(A) are assigned to events A according to the probability axioms.

[edit] Stubbification

From looking at the history of this article, one infers that once upon a time, there was a lot of important content on this page, but most of it is gone in a direction unknown, and the current version is little better than a stub with a long list of links. What an extraordinary loss! I don't know which Einstein, or, given the context, Laplace came up with the idea of removing most content from this page to who-knows-where, but in my opinion, it amounts to an act of extraordinary vandalism. It may not all happened at once, but I do not have sufficient time and resources to investigate. If any qualified editors are monitoring this page, can you, please, restore some reasonable verion of it? For one possible model of what Probability theory article can look like, you may refer to the article in the German edition of Wikipedia. Arcfrk 03:15, 21 March 2007 (UTC)

Looking at the revison just before the stubification [2] it looks like it was a rather abstract page with formal definitions beyond non specalists. These quite rightly seem to have gone to Probability space and Probability axioms. For the vital article we probably want Probability instead and I've change WP:VA accordingly. --Salix alba (talk) 09:55, 21 March 2007 (UTC)
See also Talk:Probability#Merge Probability theory to here.  --LambiamTalk 10:33, 21 March 2007 (UTC)

I strongly object both to the removal of Probability theory from the list of vital articles and to merging it with Probability. As an emergency measure, I will put links to Probability space and Probability axioms in the lead. However, probability theory is one of the most important mathematical disciplines, with enormous applications to exact sciences, social sciences, non-sciences such as economics, and through Statistics, for which it provides mathematical underpinning, to nearly every subject dealing with analysis of large amounts of data. To merge the article on Probability theory with the article on Probability is akin to merging Differential equations with Derivative. Needless to say, probability theory constitutes a lot more than a list of axioms for probality spaces. Now, to address your complaint about abstractness and being beyond non specialists: you should expect an article in an encyclopaedia to reflect the current state of knowledge, which will be beyond most people's basic training. I am not advocating for making it incomprehensible on purpose, but merely point out that the popularity of a topic among the masses cannot be a valid criterion for inclusion or exclusion it into encyclopaedia or designating it as a vital topic. If the article was incomplete or bad in some ways, it would have to be improved, not dissipated. I think that the section on probability in mathematics in Probability can form a useful core for this article. Of course, it will have to be eventually expanded and structured further. Arcfrk 00:25, 22 March 2007 (UTC)

Arcfrk, instead of pontificating on the talk page, why don't you put an outline into the article? There's no need to call people vandals or "einsteins." It's my fault that article is blank, and the reason is that I didn't know what should go here, but I was pretty sure that what was here before didn't belong. You're right about the stuff in mathematical probability being a good start to this page, maybe you'd like to build a more complete outline? --MisterSheik 00:34, 22 March 2007 (UTC)
Well, better to pontificate once and develop a good coordinated plan than engage in a meaningless game of tug-of-edit! For the future reference, you may consider writing up a motivated summary of your substantial edits, otherwise to us mere humans it may appear to be a work of an Einstein (meaning, a genius). Arcfrk 03:56, 22 March 2007 (UTC)

As I've said elsewhere, I think that it's the Probability page that should be the page without real contents. "Probability" and "probable" in natural languages can mean quite a lot of different things, only seldomly coinciding with the mathematical concept. For this reason I think that Probability should be nothing more than a disambiguation page for all the different flavors of the word. I agree that Probability theory deserves an article of its own, not only a stub. iNic 02:51, 22 March 2007 (UTC) And I saw now that MisterSheik already started to implement this idea. :-) iNic 02:57, 22 March 2007 (UTC)

It wasn't my idea--it was yours! :) But, yeah, it's a start I agree... --MisterSheik 03:25, 22 March 2007 (UTC)

[edit] New lead

I've written a new lead to this article. As you can see, it is quite different in intent and flavor from the lead in Probability. I invite experts in probability to develop the article to a level comparable with, say, Algebra or Geometry. I reiterate my belief that the mathematical theory of probability should be treated separately from a more leisurely article explaining the history and main concepts in lay terms. Arcfrk 04:01, 22 March 2007 (UTC)

It's really well written :) Nice job. --MisterSheik 04:22, 22 March 2007 (UTC)

[edit] Link to Britannica

I've restored the link to Britannica, for the following reason: the one sentence definition of probability theory is a direct quote from the Britannica article. It would seem unwise not to acknowledge this fact, and has nothing to do with being or not being freely available to everyone (think of references to books that are not "freely available" to anyone with internet access). Arcfrk 21:49, 23 March 2007 (UTC)

OK, but to have a link that supposedly goes to a source article but is blocked for most users is annoying I think. To avoid this, and still keep the reference intact, I suggest that we remove the link part but leave the reference as it is. iNic 03:05, 25 March 2007 (UTC)
Changed the Britannica link to free summary accessible by all users. Hirak 99 22:33, 28 March 2007 (UTC)

[edit] veracity

Is this even true? "The general study of probability has two main flavors - discrete and continuous."

What about a random variable that's dirichlet distributed? It is not continuous, but has a pdf.

This statement needs to be more precise. For example, it could say: "There are three theories of probability: discrete, continuous, and ...?" But, I have no idea...

Also, we should be careful not to duplicate the information that is on other pages, or belongs on other pages. For example, probability space, probability axioms, and probability distribution should all be well-written clear pages that we can just refer to here. (And if they're not, they should be fixed.) There's no point in redefining what a probability distribution is in more than a sentence or two, imho.

MisterSheik 22:28, 28 March 2007 (UTC)

Certainly the Dirichlet distribution is continuous.
Yeah :) What I meant was that it doesn't fit under the definition of continuous distributions that is on the page right now, because a dirichlet random variable is not a real number; you can't define Pr(X<4), etc. So, something is wrong with the definition on this page. (On second look, it's now dealt with by the "measure theoretic probability theory".
It is true that some probability distributions are neither discrete nor continuous (in the sense of having a pdf) nor a mixture of the two. But I'm not sure that's another "main flavor of the general study of probability". Also, I definitely would NOT say "three theories of probability" or "two theories of probability" or the like when writing about this sort of thing. They're really not separate theories. Michael Hardy 22:34, 28 March 2007 (UTC)
If there is a dichotomy in probability theory, I feel that it's between the study of independent random variables and stochastic processes. A sequence of i.i.d. variables can of course be interpreted as a stochastic process with discrete time; however, in general theory of stochastic processes many types of "somewhat dependent" processes are considered. Arcfrk 23:14, 28 March 2007 (UTC)
I have added measure theoretic probability, and changed the word "classification" to "treatment". Hopefully this will take care of all confusions. Cheers --Hirak 99 23:16, 28 March 2007 (UTC)
Yeah, it looks really good right now. Very nicely presented...MisterSheik 01:07, 29 March 2007 (UTC)
I have to confess that I'm very pleased with the article in its current state. :-) Well done! iNic 20:49, 29 March 2007 (UTC)
Thank you :) Hirak 99 14:13, 30 March 2007 (UTC)

[edit] Generating functions

Would another section on generating functions make sense? Other than the pdf/pmf other ways to characterize a real-valued rv are the... moment-generating function, characteristic function, and cumulant generating function, and explain in a couple lines what is going on? MisterSheik 01:30, 29 March 2007 (UTC)

Yes, the generating functions section seems to have a discontinuity after the Treatment section. I am thinking about bringing back a distribution section, with the mgf, cf etc in a subsection of it along with the laws like Levy continuity theorem. Also how about removing "Probability in Mathematics" and instead adding sections like law of large numbers? --Hirak 99 07:55, 29 March 2007 (UTC)
PS. Thanks for all the editing to give the article a much higher quality :) --Hirak 99 07:56, 29 March 2007 (UTC)
Good idea Hirak. I like the depth that you chose for the law of large numbers. Ideally, since the introduction also mentions the central limit theorem, that could have a similar overview-like section? But, on a completely selfish note, I don't understand how the mgf and cf and cgf relate to one other (if at all?), and I'd be thrilled to see an equally good overview here :) Maybe I should just read those articles though ;) Cheers. MisterSheik 16:19, 29 March 2007 (UTC)

[edit] Now almost totally redundant, unless someone wants to merge something back in

To give a mathematical meaning to probability, consider flipping a "fair" coin. Intuitively, the probability that heads will come up on any given coin toss is "obviously" 50%; but this statement alone lacks mathematical rigor. Certainly, while we might expect that flipping such a coin 10 times will yield 5 heads and 5 tails, there is no guarantee that this will occur; it is possible, for example, to flip 10 heads in a row. What then does the number "50%" mean in this context?

One approach is to use the law of large numbers. In this case, we assume that we can perform any number of coin flips, with each coin flip being independent—that is to say, the outcome of each coin flip is unaffected by previous coin flips. If we perform N trials (coin flips), and let NH be the number of times the coin lands heads, then we can, for any N, consider the ratio N_H \over N.

As N gets larger and larger, we expect that in our example the ratio N_H \over N will get closer and closer to 1/2. This allows us to "define" the probability \Pr(H) of flipping heads as the limit, as N approaches infinity, of this sequence of ratios:

\Pr(H) = \lim_{N \to \infty}{N_H \over N}

In actual practice, of course, we cannot flip a coin an infinite number of times; so in general, this formula most accurately applies to situations in which we have already assigned an a priori probability to a particular outcome (in this case, our assumption that the coin was a "fair" coin). The law of large numbers then says that, given Pr(H), and any arbitrarily small number ε, there exists some number n such that for all N > n,

\left| \Pr(H) - {N_H \over N}\right| < \epsilon

In other words, by saying that "the probability of heads is 1/2", we mean that if we flip our coin often enough, eventually the number of heads over the number of total flips will become arbitrarily close to 1/2; and will then stay at least as close to 1/2 for as long as we keep performing additional coin flips.

Note that a proper definition requires measure theory, which provides means to cancel out those cases where the above limit does not provide the "right" result (or is even undefined) by showing that those cases have a measure of zero.

The a priori aspect of this approach to probability is sometimes troubling when applied to real world situations. For example, in the play Rosencrantz & Guildenstern Are Dead by Tom Stoppard, a character flips a coin which keeps coming up heads over and over again, a hundred times. He can't decide whether this is just a random event—after all, it is possible (although unlikely) that a fair coin would give this result—or whether his assumption that the coin is fair is at fault.

[edit] I'm happy

I'm very pleased with the current version of this article now... it has become atleast a good base to start with.

Things that remain to do:

  1. Cleanup of the see also (med-low priority)
  2. Assembling intros other important areas in probability theory in this page (low priority - already comprehensive encyclopedia article, although surely more details can be added)

Cheers, Hirak 99 21:05, 30 March 2007 (UTC)

[edit] P and Pr

I just discovered that "P" and "Pr" aren't used consistently in the article. I would personally prefer to see P everywhere instead of Pr, but it seems that Pr is adopted as the standard notation in other articles at wikipedia. And standards are always good, aren't they? ;-) iNic 01:12, 31 March 2007 (UTC)
I guess the reason people might prefer "Pr" is that the TeX markup adopted for Wikipedia has a standard function \Pr which comes in non-italics and looks like an operator when rendered. I see no other reason for preferring "Pr". I think it is alright if one wishes to replace all \Pr in the text with \operatorname{P}, but just to follow standards, we might keep it as Pr :-) --Hirak 99 09:29, 31 March 2007 (UTC)
Yes, you might be right here. I bet there are some, maybe heated, discussions about this somewhere on wikipedia talkpages; it's the kind of minor issues that tend to start a debate. I would have voted for P but I apparently missed the voting opportunity. And if they ended up settling for Pr for some reason I will not object to that. :-) iNic 20:21, 31 March 2007 (UTC)
The confusing thing about random variables is the meaning of operations on them: sometimes we mean to operate on the random variable's value and other times we are operating on its probability distribution in a more complicated way. For example, if X is a random variable, then 2X means doubling the values of the random variable, and similarly sin(X) means applying the sin function to values. But, E(X), for example isn't a function on the values, but rather takes the whole distribution and returns a value. Similarly, P(X=2) doesn't seem like a function on the event (until you are doing measure theoretic probability theory, and you think of P as a measure on the space of events, right?) And so, I think that Pr makes more sense unless you specifically mean the measure-theoretic definition, and then you would ideally always pass it a set events. Unfortunately, no one does that, and we all use P(X=2) as shorthand. I think this is why people write E[X] (with square brackets) and Pr(X=2): to distinguish, by means of notation, these operation from run-of-the-mill functions. (But I'm not opposed to P(X=2); I just wish that probability theorists had come up with nicer notation). MisterSheik 04:23, 1 April 2007 (UTC)
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