Talk:Expected value

Archived discussion to mid 2009

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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]

Removal of citation needed rationale

I tried and failed to find citations supporting for Pascal's state of mind:

Pascal, being a mathematician, was provoked and determined to solve the problem once and for all.

I decided to change it to a more straightforward assertion:

Pascal, being a mathematician, decided to work on a solution to the problem

I did not add a citation to the paragraph as the subsequent paragraph, which recounts his work on the problem, is sourced.

Should someone track down sourcing for Pascal state of mind feel free to restore the original statement with a citation. S Philbrick (Talk) 14:58, 1 October 2025 (UTC)[reply]

Cauchy mean - undefined or indeterminate

I was initially jarred when I saw the expected value of the Cauchy distribution was stated to be "undefined". (See the table of expected values of common distributions). I see the formula for the expected value (which I haven't tried to corroborate), which in my view is a definition. However the formula doesn't converge nicely, so I've always thought of the expected value as being "indeterminate". My view was bolstered by the fact that the term "undefined" is wiki linked and brings you to Indeterminate form.

I did some superficial research on the distinction between indeterminate and undefined, especially in the context of the Cauchy distribution, and it appears my view is not supported by a preponderance of sources, so I have chosen not to change "undefined" to "indeterminate", but am leaving this comment in case someone with more bandwidth is willing to explore it in more detail. S Philbrick (Talk) 15:18, 1 October 2025 (UTC)[reply]