Talk:Expected value

Archived discussion to mid 2009

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Integration over extended real line

Part of the wiki page used integration over the extended real line, which is unusual. I see that this was previously discussed here in 2017. I have edited to give it the usual formulation which appears in textbooks. Gumshoe2 (talk) 05:04, 5 February 2022 (UTC)[reply]

If $P(X\in \mathbb {R} )=0$ and $P(X=+\infty )=1,$ then $F(x)=0$ on the entire real line and at $-\infty ,$ and $F(+\infty )=1.$ But

\int _{\mathbb {R} }x\,dF=0\neq EX=+\infty .

StrokeOfMidnight (talk) 05:54, 5 February 2022 (UTC)[reply]

For one thing, as far as I can see there is no indication given on the page of random variables taking on infinite values (and certainly not on a set of positive measure). Gumshoe2 (talk) 06:02, 5 February 2022 (UTC)[reply]

Just taking the two standard textbook references nearest to me at the moment (the probability books by Durrett and by Kallenberg), they both define random variables to be valued in the real line. Gumshoe2 (talk) 06:09, 5 February 2022 (UTC)[reply]

Furthermore I am unaware of standard references in which Lebesgue-Stieltjes integration is done on the extended real line. I can understand your preferences for how to formulate the theory, and to some extent I even share them, but this is not a good website to develop your personal preferences. Gumshoe2 (talk) 06:51, 5 February 2022 (UTC)[reply]

I have reverted, given the lack of response. Gumshoe2 (talk) 05:13, 7 February 2022 (UTC)[reply]

I did, in fact, respond: the CDF-based formula breaks when the r.v. assumes infinite values w/non-zero probability StrokeOfMidnight (talk) 05:58, 7 February 2022 (UTC)[reply]

As I said above: in standard textbooks random variables in this context are always taken to be real-valued. I see that five years ago Borel Tsirelson raised exactly the same point as me on this talk page. Following those discussions, you changed to what I am suggesting, until you changed it back in 2020. Why? Gumshoe2 (talk) 06:15, 7 February 2022 (UTC)[reply]

Looks like you are correct. Anything measurable taking its values in ${\overline {\mathbb {R} }}$ is a random element rather than a random variable. StrokeOfMidnight (talk) 07:40, 7 February 2022 (UTC)[reply]

"Absolutely continuous random variables" vs "continuous random variables" in section title

Gumshoe2, may I ask why you keep changing the section title from "Absolutely continuous random variables" to "Continuous random variables" without explanation? While both naming conventions are common, the former is more accurate since the section discusses exclusively random variables with densities. Given that the section has a very narrow goal (which is to define the expected value), choosing an accurate title is important, and the title you seem to insist on is ambiguous, as it describes two different concepts simultaneously. StrokeOfMidnight (talk) 10:54, 8 February 2022 (UTC)[reply]

I apologize, I thought I had made it clear. I am not just changing the title, I am also changing the content, as addressed in my above comments in section "The mathematical level of writing". The notion of absolutely continuous random variables is measure-theoretic and so only appropriate in the following section. I recognize that "continuous random variable" is ambiguous, although as far as I can see there are (at the moment) zero ambiguities in the (non-title) text itself. I have now made change to "Random variables with piecewise-continuous density", which should be fully accurate. Gumshoe2 (talk) 14:27, 8 February 2022 (UTC)[reply]

There is actually some similar issue with "Arbitrary random variables" since the random variables are of course not totally arbitrary. I still think "Measure-theoretic random variables" is accurate and clear, although if I recall correctly, you say there is no such thing as a measure-theoretic random variable. So I am not sure what is best in this case. Gumshoe2 (talk) 14:30, 8 February 2022 (UTC)[reply]

The new title is definitely better BUT:

PDF is not required to be piecewise continuous;
"piecewise" in itself is a deceptive term: can the discontinuity set be countably-infinite?
finding a reasonable PDF, say on $[0,1],$ with countably many discontinuities is straightforward;
removing p.w.c. creates a shorter title that is just as informative and does not involve an extraneous concept. StrokeOfMidnight (talk) 23:05, 8 February 2022 (UTC)[reply]

Yes, all that you mention is (implicitly or explicitly) contained in the measure-theoretic subsequent section presently called "Arbitrary random variables". Gumshoe2 (talk) 23:09, 8 February 2022 (UTC)[reply]

I have to admit I don't actually understand what your complaint is at present, given that the content on general densities that you want is already present on the page, in the next section. Why can this particular section not give an elementary reductive case, just like the two previous sections on discrete random variables? Gumshoe2 (talk) 23:16, 8 February 2022 (UTC)[reply]

Yes, I understand what you mean by "reductive case". My point was: p.w.c. doesn't do anything useful here. StrokeOfMidnight (talk) 23:18, 8 February 2022 (UTC)[reply]

If you were teaching a course on measure theory, or if this were a wiki page on measure theory, it would indeed be a strange and useless condition to add. The usefulness of p.w.c. here is that (1) it is present in very standard textbooks (the given ref is maybe the most standard probability ref possible); (2) it is mathematically precise (moreso than your revision to just saying that the density is "a function"); (3) it is understandable to a wider variety of people than the alternative "general" density, which requires understanding of measurable functions. Do you disagree with any of these three points? Gumshoe2 (talk) 23:27, 8 February 2022 (UTC)[reply]

On point 1: that's far from universal. P.w.c. is present in some texts, absent in others.
On point 2: why do you need this precision when it doesn't simplify anything? Same formulas, same everything.
On point 3: at this point in the article, one needs to, at least, have seen the definition of PDF. There is just no way around this.
I would be surprised if anybody even consciously thinks of p.w.c functions in this context unless you specifically mention them. StrokeOfMidnight (talk) 23:58, 8 February 2022 (UTC)[reply]

(1) I agree. If you have a better suggestion for a simple condition which is common in textbooks, I think it would be ok to use instead. (2) Perhaps I don't understand what you mean by the question, since you say have a Ph.D. in math, so presumably you understand well the importance of precision whenever it is possible. (3) It is absolutely not necessary to understand measure theory at this point, as a very large number of people (probably a strong majority) have an understanding of density functions and probability without understanding measure theory. Gumshoe2 (talk) 00:06, 9 February 2022 (UTC)[reply]

1: The suggestion is already there and has been for a long time (long before this article was written): let the r.v. have a density function. Then the e.v. may be expressed as an integral. (Riemann integral, in most cases). That's a real benefit. This is exactly the "reductive case" you were talking about.
2: I mean that p.w.c. is an extraneous concept and brings no additional benefit: same formulas, same everything.
3: Agreed. However, it is necessary to know the basics including (some stripped-down form of) probability measure and density function. Density functions are non-negative and integrate to 1. No one can strip that down further. Nor should one. StrokeOfMidnight (talk) 00:54, 9 February 2022 (UTC)[reply]

I find your perspective baffling, but it seems we are unlikely to come to a common understanding of the matter any time soon. I hope this new edit will satisfy us both? Gumshoe2 (talk) 02:53, 9 February 2022 (UTC)[reply]

Conceptually, the edit looks ok, although the wording is a bit convoluted, IMO. StrokeOfMidnight (talk) 03:06, 9 February 2022 (UTC)[reply]

I changed the wording and broke up a long sentence. StrokeOfMidnight (talk) 03:13, 9 February 2022 (UTC)[reply]

Ok, great. Your rewrite does read better. I have reworded a little extra, and added back in the fact (which we seem to agree to be a fact!) that the theory is often done for piecewise-continuous functions - I regard this as very important to say. I am satisfied with the section for now. Gumshoe2 (talk) 03:27, 9 February 2022 (UTC)[reply]

One remaining issue: we should still mention the term "absolutely continuous" briefly. It wouldn't make sense to discuss them without naming them. StrokeOfMidnight (talk) 04:05, 9 February 2022 (UTC)[reply]

Ok, done. Gumshoe2 (talk) 04:09, 9 February 2022 (UTC)[reply]

Sorry, I didn't notice you had already done so a minute before. (However I think my version is preferable since it is integrated into the text.) Gumshoe2 (talk) 04:18, 9 February 2022 (UTC)[reply]

I have rephrased it now, hopefully ok? Gumshoe2 (talk) 04:29, 9 February 2022 (UTC)[reply]

Since "continuous random variables" are referred to everywhere in the literature, I have added one extra sentence giving one possible definition and clarifying that they are not consistently defined. I think this is necessary so as to be encyclopedic. Gumshoe2 (talk) 04:16, 9 February 2022 (UTC)[reply]

Looks good. StrokeOfMidnight (talk) 05:35, 9 February 2022 (UTC)[reply]

Great! Gumshoe2 (talk) 05:37, 9 February 2022 (UTC)[reply]

Deleted history section

I have deleted the history section, since aside from being (in my opinion) unclearly written, it is taken word for word from Cain Mckay's book "Probability and statistics". I have left the one remaining sentence. (I don't think it would be any loss to lose also.) Gumshoe2 (talk) 04:03, 10 February 2022 (UTC)[reply]

I just realized that the plagiarism must be the other way around, since the book was published in 2019, after the text was added here. My mistake! Gumshoe2 (talk) 04:08, 10 February 2022 (UTC)[reply]

Anyway, if anyone has secondary sources for this material, it would be very helpful. For instance, I am very confused by the etymology section, since it seem to not be about etymology, and it seems like Huygens is using expectation in the modern sense. And it seems like Laplace is defining the summands of expectation, not expectation. Also, I am skeptical of the Whitworth reference and the reliability of the "Earliest uses of symbols in probability and statistics" webpage cited; as I mentioned in an edit summary, the fourth edition of Whitworth's book used E, which is from four years earlier than the year (1901) given here. So the claim seems to be untrue. Gumshoe2 (talk) 04:13, 10 February 2022 (UTC)[reply]

Equity

In games of chance, such as poker or backgammon, the "expected value" is commonly referred to as the "equity" or "fair market value" of a position. For instance this definition:

Equity: The value of a position to one of the players. Equity is the sum of the values of the possible outcomes from a given position with each value multiplied by its probability of occurrence. It is the same as the fair settlement value of the position. Your equity is the negative of your opponent's equity.

But I'm not seeing the term used in this article. There is a section about equity in the poker strategy article, but it's unsourced and could otherwise stand some improvement. It links to this article, but this article has little follow up for those readers. Other than that, I'm not seeing a good treatment of equity (or expected value) in simple games of chance anywhere on Wikipedia, although many articles such as craps, casino, and board game link to this article. I think that's an omission that should be corrected.

Is this article the best place for it? Does it belong somewhere else? This article is fairly technical and seems to assume that the reader is familiar with calculus, perhaps even measure theory, so maybe somewhere else would make more sense.

Seems to me that a simple exposition on expected value for finite sample spaces, along with its interpretation as "equity" or "fair market value" and three or four simple examples would improve things. This simple exposition could be accomplished with only arithmetic (addition, multiplication, division) which would make it more accessible to the general reader. BTW, in the current two examples, a single die roll and roulette, the sample space has equal probability for all outcomes. It would be more illustrative to have an example where the probabilities were different to demonstrate weighted average. Mr. Swordfish (talk) 15:23, 4 April 2023 (UTC)[reply]

Too much about density functions

The two subsections

of the section Definition currently deal with many details on density functions and absolutely continuous random variables. However, the appropriate places for these are the two articles Random variable and Probability density function. I therefore suggest deleting a lot of details on density and absolute continuity here or moving it there. Rigormath (talk) 11:37, 18 May 2024 (UTC)[reply]

As a mathematician, I would agree with you if the purpose were to efficiently define the expected value in the greatest generality. But the expected value via densities is absolutely ubiquitous in standard sources, so it has to be given its due weight. Gumshoe2 (talk) 13:39, 18 May 2024 (UTC)[reply]

The following red excerpt, for example, would be helpful in the article Probability density function, but does not contribute to the understanding of the present definition and its "due weight":

However, the Lebesgue theory clarifies the scope of the theory of probability density functions. A random variable

X

is said to be absolutely continuous if any of the following conditions are satisfied:

there is a nonnegative measurable function $f$ on the real line such that

{\text{P}}(X\in A)=\int _{A}f(x)\,dx,

for any Borel set

A

, in which the integral is Lebesgue.

the cumulative distribution function of $X$ is absolutely continuous.
for any Borel set $A$ of real numbers with Lebesgue measure equal to zero, the probability of $X$ being valued in $A$ is also equal to zero
for any positive number $ε$ there is a positive number $δ$ such that: if $A$ is a Borel set with Lebesgue measure less than $δ$ , then the probability of $X$ being valued in $A$ is less than $ε$ .

These conditions are all equivalent, although this is nontrivial to establish.^[20] In this definition,

f

is called the probability density function of

X

(relative to Lebesgue measure).

By the way, don't underestimate the readers' ability to klick a link to another article. Rigormath (talk) 15:17, 19 May 2024 (UTC)[reply]