Talk:Expected value

Archived discussion to mid 2009

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)

Too much about density functions

The two subsections

• Random variables with density
• Arbitrary real-valued random variables

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)

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)

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)

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)

Pascal and Gombaud had a heated philosophical discussion about the meaning and usefulness of mathematics in general and about some mathematical concepts in particular. Later in Pascal's letters to Fermat it shows how unhappy, to say the least, Pascal was about Monsieur Gombauds philosophical stance about mathematics. He talked about Gombaud in a derogative manner saying things like "Monsieur Gombaud is very clever but not a mathematician which, as you know, is a big flaw." The two problems Gombaud brought up when they met was not just two randomly chosen problems, but two problems he thought was showing exactly why his philosophical stance on the matter was correct. Pascal didn't share his views at all. On the contrary it was very important for Pascal that the world could be described using mathematics. To claim, as Gombaud did, that mathematics is poorly connected to reality, and even self-contradictory, was very provoking for Pascal. As it would have been to almost any mathematician, both then and now. Pascals urge to win this philosophical dispute, or at least show that the arguments Gombaud had put forth for his stance wasn't valid arguments, he decided to show that these two problems indeed could be solved mathematically, without contradictions. This was the reason he started to write letters to Fermat about this. These letters (those that remain) can be found online if you want to read them. Fermat viewed the two problems more like two randomly chosen exercises that he solved with ease, while for Pascal it was much more emotional. He was a deeply religious person (even for his time) and that his solution of the problem of points coincided with Fermat's result made a huge emotional impact on him. He interpreted his and Fermat's result as the formula to use as a moral guide in life in general. Later in one of his religious books he turned the formula into his famous argument for why one should believe in a god, today called Pascal's Wager. So to say that Pascal got PROVOKED by the philosophical views Monsieur Gombaud expressed, regarding the utility of mathematics, is hardly an overstatement. 'Anger' or 'holy rage' would probably be closer to the truth. iNic (talk) 23:58, 6 February 2026 (UTC)

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)