Talk:Normal distribution

Removed Section

Below you will find the removed section with additional comments for you to review. Added more comments for basic mathematical operations. Later you are free to remove them if you think that they are not worth mentioning. Furthermore I would support to add layman explanations (see comments above). --Bert Niehaus (talk) 15:03, 6 February 2026 (UTC)

Cumulative Distribution Function

With the following probability density function

${\displaystyle {\begin{array}{rcl}f(x\mid \mu ,\sigma ^{2})&=&\displaystyle {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}},\ \ \ -\infty <x<+\infty \\&=&\displaystyle {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot \underbrace {\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!\cdot (2\cdot \sigma ^{2})^{n}}}\cdot (x-\mu )^{2n}} _{=e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}\\\end{array}}}$

the normal distribution has the following cumulative distribution function (see taylor series in complex analysis):

${\displaystyle {\begin{array}{rcl}F(x)&=&\displaystyle {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\int _{-\infty }^{x}e^{-{\frac {1}{2}}\left({\frac {t-\mu }{\sigma }}\right)^{2}}\mathrm {d} t,\quad x\in \mathbb {R} \\&=&\displaystyle {\frac {1}{2}}+{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!\cdot (2\cdot \sigma ^{2})^{n}}}\cdot {\frac {(x-\mu )^{2n+1}}{2n+1}}\\&=&\displaystyle {\frac {1}{2}}+{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot \sum _{n=0}^{\infty }{\frac {1}{n!\cdot (2n+1)}}\cdot \left({\frac {-1}{2\cdot \sigma ^{2}}}\right)^{n}\cdot (x-\mu )^{2n+1}\\\end{array}}}$

Taylor Series - Complex Analysis

The density of the standard normal distribution extended to the complex numbers ${\displaystyle \mathbb {C} }$ can be represented directly by the composition of two entire functions ${\displaystyle f(z)=\exp(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}}$ and ${\displaystyle g(z)=-{\frac {z^{2}}{2}}}$. The composition ${\displaystyle f\circ g=\varphi }$ together with the taylor series for the exponential function represents ${\displaystyle \varphi }$. Furthermore ${\displaystyle \varphi }$ is as composition of two entire functions again an entire function.

${\displaystyle {\begin{array}{rcl}\varphi (z)&=&\displaystyle {\frac {1}{\sqrt {2\pi }}}\cdot e^{-{\frac {z^{2}}{2}}}={\frac {1}{\sqrt {2\pi }}}\cdot \sum _{n=0}^{\infty }{\frac {\left(-{\frac {z^{2}}{2}}\right)^{n}}{n!}}\\&=&\displaystyle {\frac {1}{\sqrt {2\pi }}}\cdot \sum _{n=0}^{\infty }{\frac {(-1)^{n}\cdot {\frac {z^{2n}}{2^{n}}}}{n!}}\\&=&\displaystyle {\frac {1}{\sqrt {2\pi }}}\cdot \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!\cdot 2^{n}}}\cdot z^{2n}\end{array}}}$

The extension of ${\displaystyle \varphi$ :\mathbb {R} \to \mathbb {R} } to ${\displaystyle \varphi$ :\mathbb {C} \to \mathbb {C} } is done to use the results from complex analysis. The identity theorem shows that the taylor series is unique for ${\displaystyle \mathbb {C} }$ with:

${\displaystyle {\begin{array}{rrcl}\varphi$ :&\mathbb {C} &\rightarrow &\mathbb {C} \\&z&\mapsto &\displaystyle \varphi (z)={\frac {1}{\sqrt {2\pi }}}\cdot e^{-{\frac {z^{2}}{2}}}={\frac {1}{\sqrt {2\pi }}}\cdot \underbrace {\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!\cdot 2^{n}}}\cdot z^{2n}} _{=e^{-{\frac {z^{2}}{2}}}}\end{array}}}

With ${\displaystyle \mathbb {R} \subset \mathbb {C} }$ it provides also the taylor series on ${\displaystyle \mathbb {R} }$ with the radius of convergence ${\displaystyle +\infty }$. On a closed disk ${\displaystyle D_{r}(0):=\{z\in \mathbb {C} \,:\,|z|\leq r\}}$ the partial sums of the taylor series converges uniformly against ${\displaystyle \varphi }$ and this provides the antiderivative of ${\displaystyle \varphi }$.

${\displaystyle {\begin{array}{rrcl}{\widehat {\Phi }}_{r}:&{\overline {D_{r}(0)}}&\rightarrow &\mathbb {C} \\&x&\mapsto &\displaystyle {\widehat {\Phi }}_{r}(x)={\frac {1}{\sqrt {2\pi }}}\cdot \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!\cdot 2^{n}}}\cdot {\frac {1}{2n+1}}\cdot z^{2n+1}\end{array}}}$

Identity theorem in complex analysis provides again a unique extension of ${\displaystyle {\widehat {\Phi }}_{r}}$ to ${\displaystyle {\widehat {\Phi }}:\mathbb {C} \to \mathbb {C} }$. In a last step symmetry of ${\displaystyle {\widehat {\Phi }}}$ consideration and the properties of a distribution function lead to the Taylor series for ${\displaystyle \Phi$ :\mathbb {R} \to [0,1]} with:

${\displaystyle \Phi (x)={\frac {1}{2}}+{\widehat {\Phi }}(x)={\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\cdot \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!\cdot 2^{n}}}\cdot {\frac {1}{2n+1}}\cdot x^{2n+1}}$

The ${\displaystyle {\frac {1}{2}}+...}$ is necessary because a cumulative distribution function needs to fulfill:

${\displaystyle \lim _{x\to -\infty }\Phi (x)=0{\mbox{ and }}\lim _{x\to +\infty }\Phi (x)=1}$

Without the ${\displaystyle {\frac {1}{2}}+...}$ the limits are:

${\displaystyle \lim _{x\to -\infty }{\widehat {\Phi }}(x)=-{\frac {1}{2}}{\mbox{ and }}\lim _{x\to +\infty }{\widehat {\Phi }}(x)=+{\frac {1}{2}}}$

This violates the required properties of a cumulative distribution function.

Linear Transformation

With an expectation value ${\displaystyle \mu \in \mathbb {R} }$ and the variance ${\displaystyle \sigma ^{2}>0}$ a linear transformation of ${\displaystyle \Phi }$ provides the Taylor series of ${\displaystyle F}$ for an arbitrary normal distribution:

${\displaystyle {\begin{array}{rcl}F(x)&=&\Phi \left({\frac {t-\mu }{\sigma }}\right)=\displaystyle {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\int _{-\infty }^{x}e^{-{\frac {1}{2}}\left({\frac {t-\mu }{\sigma }}\right)^{2}}\mathrm {d} t,\quad x\in \mathbb {R} \\&=&\displaystyle {\frac {1}{2}}+{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!\cdot (2\cdot \sigma ^{2})^{n}}}\cdot {\frac {(x-\mu )^{2n+1}}{2n+1}}\\\end{array}}}$

The linear transformation of ${\displaystyle \varphi }$ resp. ${\displaystyle \Phi }$ provides the final cumulative distribution function mentioned above.

--Bert Niehaus (talk) 15:01, 6 February 2026 (UTC)

Thank you for the comments on my talk page.

• if ${\displaystyle \wedge }$ should not be used because a logical "and" cannot be understood by the general audience, then changing a logical correct ${\displaystyle \wedge }$ to a more comprehensive "and" would also be an option for a community based approach like Wikipedia.
• will change that in the comments above.
• If something is reverted, I thought I should provide the basic mathematical arguments on the talk page, so the community can understand why the Taylor series is just a basic mathematical composition of two entire functions as mentioned above.
• I just did a substitution of ${\displaystyle -{\frac {x^{2}}{2}}}$ in the Taylor series of the exponential function ${\displaystyle exp(x)=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}$. The definition of the probability density was there before my edit.
• I will change the ${\displaystyle \wedge }$ to an "and" on the talk page as recommended.

Bert Niehaus (talk) 10:11, 9 February 2026 (UTC)

• Recommendation: I would recommend to the community to add at least

${\displaystyle {\begin{array}{rcl}f(x\mid \mu ,\sigma ^{2})&=&\displaystyle {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}},\ \ \ -\infty <x<+\infty \\&=&\displaystyle {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot \sum _{n=0}^{\infty }{\frac {1}{n!}}\cdot \left({\frac {-1}{2\cdot \sigma ^{2}}}\right)^{n}\cdot (x-\mu )^{2n}\\\end{array}}}$

so that the reader is able to understand, why the cumulative distribution function ${\displaystyle F}$ has the mentioned Tailor series. A short sentence like

"As a composition of 2 entire functions with domain ${\displaystyle \mathbb {C} }$ the density function has an antiderivative on ${\displaystyle \mathbb {C} }$ (and on ${\displaystyle \mathbb {R} )}$."

could support the reader, if mentioning the identity theorem and uniform convergence would be too much information for the reader. Nevertheless I would encourage you to mention that as well. It is one sentence and it helps at least mathematicians to understand why the Taylor series of the cumulative distribution functions must look like the one mentioned here in the talk page. I apologize that the community must revert two times my contributions. All the best, Bert--Bert Niehaus (talk) 15:22, 9 February 2026 (UTC)

Dear @Bert Niehaus,

Re. "I apologize that the community must revert two times my contributions." → please do not worry about it. Your contributions were obviously made in goodfaith and were, from a purely mathematical stand point, of high quality. You did not introduce anything incorrect in the article, so there is absolutely no harm done. And, having in mind the countless instances of vandalism that have to be reverted every single minute, it really didn't make any difference to anyone. I just hope you bear me no ill-will for reverting your edits.

I agree that it would be a good idea to mention that the power series expansion of the CDF can be obtained by integrating the power series for the PDF term by term. I don't think we need to go into details to detail the calculation or even justify why it works. However, I think that what would be great would be to provide the reader with a precise reference to a textbook where this is done. I can't think of one off the top of my head, but I guess it should be to hard to find one...

(if we can't find one, then unfortunately from the point of view of Wikipedia we will be on the border between WP:OR and WP:CALC — you and I would probably consider that it's OK because of WP:CALC; but I think many editors who are less familiar with the topic could legitimately disagree and argue that this is WP:OR.

Cheers, Malparti (talk) 17:02, 9 February 2026 (UTC)

References - Example'

Fischer/Lieb "Course in complex analysis" ^[1] has 250+ pages but on the beginning page 1-18 (including uniform convergence) is relevant as citation. It contains also a proof for the identity theorem but recommend to add additional references for that topic rather to Wikipedia identity theorem than to "normal distribution". In the context of complex analysis composition ${\displaystyle f\circ g_{1}}$ of ${\displaystyle f(z)=exp(z)}$ and ${\displaystyle g_{1}(z)={\frac {1}{z}}}$ much more interesting than ${\displaystyle f(z)=exp(z)}$ and ${\displaystyle g_{2}(z)=-{\frac {z^{2}}{2}}}$ (${\displaystyle f\circ g_{2}}$ is just an entire function and ${\displaystyle f\circ g_{1}}$ has an essential singularity - see Casorati-Weierstrass). Anyhow just the beginning p.1-18 is relevant as reference for calculating the Taylor series as mentioned above. Feel free to replace that by any other reference that serves as a more comprehensive reference for the community.

Good Faith Edit '

@Malparti You mentioned "Good Faith" assumption to me in one comment on my talk page. You can expect the same from my end too, if you revert something that might be OK for the community as contribution to the article about normal distribution. The talk page exists to handle those issues. I added the mathematical arguments, so that the community has a bit more information to look at it. Wikipedia is a community driven project and the community should be able to follow the mathematical logic. If master students in mathematics have sometimes mathematical questions even if they have access to the specific text book of the lecture then it could make sense to add some additional information for comprehensive community support in Wikipedia.

Background information - Community Support'

Despite vandalism "Good Faith" assumption is the foundation of collaborative work. This discussion is valuable for a community based approach. On the other hand extending an Wikipedia article in length too much can also be challenge for an encyclopedia if someone just wants to look up a specific definition (e.g. "What is a normal distribution?"). MediaWiki allows to subpages like:

• Background information - Link: Normal_distribution/Background
• Examples Link: Normal_distribution/Examples
• ....

These subpages do not exist and I don't want to create them and somebody else must delete them later on.

These kind of subpages (if used) could keep the core article clean and short and it allows the community to access more background information that shows why encyclopedia article looks like it is.

The background information can also be expanded in a case if good faith edit must be reverted and there was e.g. a logic error in the edit (e.g add a counter example in the subpage).

REMARK: This option to "use subpages" should be discussed in the community and not used just because it mentioned here in the talk page.

Bert Niehaus (talk) 07:57, 11 February 2026 (UTC)

@Bert Niehaus Thanks for the reference suggestion; however I think you are still not completely clear regarding what Wikipedia is and is not: again, Wikipedia is very different from a textbook; it is also very different from projects such as Wikiversity or Pr∞fwiki — and I feel like so far you have been treating Wikipedia more like if were Wikiversity.

If we simplify a bit, statements on Wikipedia should be of the form "X is true, here is why", but of the form "this source says that X is true". Of course, there is a bit more to it than that, because we have to make sure we do not give WP:UNDUE weight to some assertions, that the end results is actually helpful and pleasant to read, etc. But that's the idea.

So, if you plan to keep contributing to Wikipedia, it is very important that you understand that, here, references that explain uniform convergence and the identity theorem are not relevant. What we need is a reference where the power series expansion for the cumulative distribution function is written, exactly as it is written here (possibly with minor modifications that every body can follow, such as basic arithmetic or different variable names). This is the essential first step. And, as a second-order improvement to the article, we can add a sentence such as "This expansion can be obtained by integrating the power series for the PDF term by term", with a reference to a source where this is done.

Also, a point of detail — you might already know it, but just in case: the goal of the talk pages is for editors to discuss how to improve the article, not to add extra material that would be interested to interested readers. See e.g. WP:TALK for more on this — from which, by the way, you'll see that even though I'm replying here because it's more convenient, since technically we are talking more about how Wikipedia works than how to improve this specific article, we should probably do so on your talk page.

Cheers, Malparti (talk) 10:32, 11 February 2026 (UTC)

I apologize for not carefully reading through all of the above. I'd like to second that, generally, Wikipedia is not a place for mathematical derivations. Rather, Wikipedia is a place to present results — and only if some reliable source, such as a well-established textbook, has already presented those results. If one can briefly and obviously (WP:CALC) give an explanation for a result, that can be good, but if the explanation isn't brief, that's usually a sign that it doesn't belong. As I see it, the acid test for WP:CALC and derivations is what other editors say — consensus among editors is enough to overrule the restrictions on derivations and WP:CALC. So, edit away boldly, but know that sometimes editors will object ... and revert some of your changes. —Quantling (talk | contribs) 22:30, 11 February 2026 (UTC)

@Malparti I did not create or add subpages as in Wikiversity, I just mentioned that option. Bert Niehaus (talk) 09:40, 14 February 2026 (UTC)

I reverted, yet again, your latest edits to the page. Please read WP:LEAD to understand why, but in a nutshell the lead is not the right place for this level of detail. Malparti (talk) 11:36, 14 February 2026 (UTC)

@Malparti if it is not the right place, then it could be option to move it two the right place.

• Level of detail: If I look on the article about essential singularities ${\displaystyle exp(1/z)=\sum _{n=0}^{\infty }{\frac {z^{-n}}{n!}}}$ is mentioned as essential singularity. Of course the reader should know Laurent series and poles.
• Is ${\displaystyle \exp \left(-{\frac {z^{2}}{2}}\right)}$ more difficult than ${\displaystyle exp(1/z)}$?
• exponential function has the domain ${\displaystyle \mathbb {C} }$ and polynomials have the domain ${\displaystyle \mathbb {C} }$ in complex analysis. I guess you don't want a proof for ${\displaystyle \mathbb {R} \subset \mathbb {C} }$ to use the results in ${\displaystyle \mathbb {R} }$.
• It is not OK to ignore the results in complex analysis. They belong to the mathematical knowledge similar to probability theory.
• If the theorems are available for holomorphic functions and ${\displaystyle \exp \left(-{\frac {z^{2}}{2}}\right)}$ is one, then a link to identity theorem and uniform convergence should be sufficient.

Bert Niehaus (talk) 18:15, 14 February 2026 (UTC)

To me, it looks like you are not even trying to understand what I am trying to tell you (see e.g. your latest edit where you added a reference to support a concept (uniform convergence), when I already told you that sources are supposed to support statements and I linked several help pages and guidelines.

I have invested enough of my time linking the same help pages and guidelines over and over, so I am going to ask third party input.

Malparti (talk) 13:53, 15 February 2026 (UTC)

I agree that this addition is not justified in its present form. The book being cited does not discuss the normal distribution (I can't find any uses of "normal distribution", "Gaussian distribution", etc. in it at all). We do not need a citation for what "uniform convergence" is; we need a citation that explains why doing this calculation for the normal distribution is important. Why is it meaningful that the CDF is an entire function? Who is plugging complex numbers into a function that, by definition, gives the probability that the value of a random variable is less than ${\displaystyle x}$? Does extending into the complex plane a function whose very definition and purpose depends upon its domain being an ordered field make a whit of sense? What vital theorem is proved by this maneuver? Stepwise Continuous Dysfunction (talk) 20:42, 15 February 2026 (UTC)

A theorem in mathematics is a generalized result. If all holomorphic functions can be represented locally on open disks ${\displaystyle D_{r}(x_{o})}$ into a power series, not every function must be discussed as a single case. The representation as power series does not depend on being an ordered field. The identity theorem just provides, that the representation as power series is unique on ${\displaystyle \mathbb {C} }$ (OK, not necessary here for normal distribution - too much detail - drop that). The exponential function is defined in complex analysis on ${\displaystyle \mathbb {C} }$:

$${\displaystyle e^{it}=\cos(t)+i\cdot \sin(t)\quad \quad e^{x+it}=e^{x}\cdot (\cos(t)+i\sin(t))\quad t,x\in \mathbb {R} }$$

I do not claim Euler's formula to be mine, just use ${\displaystyle t=0}$. Bert Niehaus (talk) 09:15, 17 February 2026 (UTC)

Added a reference with an older version of the cited book Fischer Lieb (2012) showing:

${\displaystyle p(z)=\sum _{n=0}^{\infty }a_{n}\cdot (z-z_{0})^{n}}$ is ${\displaystyle p'(z)=\sum _{n=1}^{\infty }n\cdot a_{n}\cdot (z-z_{0})^{n-1}}$. It is from 1988^[2]. You may want to replace that with the already mentioned english version of that text book from 2012 as reference or choose any other text book that you think to be more comprehensive. Bert Niehaus (talk) 06:46, 19 February 2026 (UTC)

@Bert Niehaus All of the material you keep adding is both unsourced and distracting. Wikipedia articles are not supposed to be either a textbook or a list of every fact anyone can think up. Instead, the steps to deciding what to include in an article, especially something very widely used like the normal distribution, should generally be (A1) search for the best topical secondary/tertiary sources you can find directly focused on the topic and read them carefully, and then (A2) summarize their content. The method you seem to be pursuing instead is more like (B1) Wikipedian thinks independently about what they personally find interesting, works it out on their own scratch paper, and adds it to the article, and then (B2) pray that someone else can hunt up a source. But that's a bad approach. –jacobolus (t) 16:59, 28 February 2026 (UTC)

A 5 minute search turns up some example sources you could start from (this is not an endorsement of these, and feel free to search for others):

• Patel (1982) Handbook of the Normal Distribution
• Bryc (1995) The Normal Distribution
• Ahsanullah, Kibria, & Shakil (2014) Normal and Student's t Distributions and Their Applications
• Lyon (2014) "Why are Normal Distributions Normal?"
• Stahl (2018) "The Evolution of the Normal Distribution"

Or you could look at the entries about the normal distribution in more general reference works like mathematical or statistical encyclopedias, the CRC Standard Mathematical Tables, Abramowitz and Stegun, or the like. Or you could look for more general books about probability and/or statistics with chapters dedicated to the normal distibution. –jacobolus (t) 19:36, 28 February 2026 (UTC)

Added Citation

The added reference Brown (2025)^[3] mentions

${\displaystyle F(b)={\frac {1}{\sqrt {2\pi }}}\int _{0}^{b}e^{-{\frac {x^{2}}{2}}}\,dx}$

as the "upper half of the cumulative distribution function (or CDF) of the standard normal". If that is correct, the cumulative distribution function would have the property ${\displaystyle F(0)=0\not ={\frac {1}{2}}}$. @Jacobulus why did you add that citation and removed the following sentence as explanation?

"By extending the domain to ${\displaystyle \mathbb {C} }$ the following power series is unique for ${\displaystyle \varphi }$ on ${\displaystyle \mathbb {C} }$ and valid on ${\displaystyle \mathbb {R} }$. The power series can be obtained directly by composition of the power series for the two entire functions ${\textstyle x\mapsto e^{x}}$ and ${\textstyle x\mapsto -x^{2}/2}$"

— Preceding unsigned comment added by Bert Niehaus (talk • contribs) 17:35, 23 February 2026 (UTC)

I don't know why they left out the extra constant ⁠${\displaystyle {\tfrac {1}{2}}}$⁠ (or they could have just left the left endpoint as ⁠${\displaystyle -\infty }$⁠). Maybe you can find another source which specifically discusses this topic? Your previous version didn't include any such sources at all. The problem I have with the previous text is that it distracting and pointlessly inaccessible, belaboring details that are irrelevant to less-technical readers, obvious to more technical readers, not backed by any cited source. We should strive to make text of wikipedia articles readable and accessible, focusing on prose explanations giving context and motivation rather than lots of jargon-filled technical asides. –jacobolus (t) 18:23, 23 February 2026 (UTC)

Abramowitz and Stegun mentions the power series, without saying more about it. –jacobolus (t) 18:36, 23 February 2026 (UTC)

• CDF is an antiderivative of the probability density ${\displaystyle \varphi }$, but not every antiderivative is a CDF of the density. There is no choice for the constant ${\displaystyle {\frac {1}{2}}}$ of the antiderivative due to the properties of a CDF (${\displaystyle \lim _{x\to \pm \infty }F(x)=...}$).
• Wikipedia is representing a knowledge graph with links, so if you remove those link to identity theorem for ${\displaystyle \varphi$ :\mathbb {C} \to \mathbb {C} } as holomorphic functions, Euler's formula, ... the reader is able to follow the links and get the details or the reader does not want to have those details, which is also OK.

Bert Niehaus (talk) 08:11, 24 February 2026 (UTC)

Yes, but if you turn every (frankly almost trivial to anyone who finished a high school calculus class) one-simple-English-sentence idea into a confusing maze of wikilinked jargon words, that doesn't really help anyone. The readers who already knew the relevant background don't need the reminder, and the readers who didn't are just pointlessly confused and directed to topics that aren't really relevant and they aren't prepared to engage with. Someone trying to find out the Taylor series for the CDF of the normal distribution is almost certainly not looking to learn about holomorphic functions or Euler's formula, both of which are basically non sequiturs in context. –jacobolus (t) 08:26, 24 February 2026 (UTC)

FWIW, the reference to Brown & Brown was removed (based on the edit summary, for an invalid reason, but I'm happy to leave the reference out, as I agree that their getting the constant wrong is inexplicable and confusing). –jacobolus (t) 08:30, 24 February 2026 (UTC)

@Jacobolus I agree that the choice made by Brown & Brown is so confusing that it is inexplicable; but to be fair it's not technically "wrong", since the expression "upper half of the cumulative distribution function" is so vague that it could be interpreted as ${\displaystyle \Phi (x)-\Phi (0)}$ for ${\displaystyle x>0}$. I'm not trying to defend their choice at all — I do find it terrible. But if someone were to tell me:

"Let's study the CDF ${\displaystyle \Phi$ :\mathbb {R} \to (0,1)} . Since ${\displaystyle \Phi (-x)=1-\Phi (x)}$, ${\displaystyle \Phi }$ is symmetric around ${\displaystyle (0,1/2)}$; so let's just focus on ${\displaystyle F:x\mapsto \Phi (x)-1/2}$ for ${\displaystyle x>0}$ and call it the 'upper half' of ${\displaystyle \Phi }$",

which I guess is what happened here (I don't have access to that article), I would say that the terminology and notation aren't very smart, because (1) the name "upper half of the CDF" will typically be interpreted as the restriction of the CDF to ${\displaystyle \mathbb {R} _{+}}$ and (2) ${\displaystyle F}$ is the standard notation for CDFs, not their "upper half". But I wouldn't say it's incorrect — more like an error in judgment. ;)

I've tried to find a "clean" reference to the Taylor expansion of the CDF online and in the probability textbooks I have at hand, but came back empty-handed... Malparti (talk) 09:58, 24 February 2026 (UTC)

The paper wasn't really directly about this. I was just citing it as a source for the explanation of how to get a Taylor series for the PDF (by using the Taylor series for exp) and then how to use that to get the CDF (integrating term by term). I imagine there are other sources stating more or less the same, but I'm not immediately finding one. (For example I found one source which says that for any symmetric distribution among some big family of distributions you can get the Taylor series for the CDF by integrating term by term and adding ⁠${\displaystyle {\tfrac {1}{2}}}$⁠, but it doesn't directly mention the normal distribution.) –jacobolus (t) 15:55, 24 February 2026 (UTC)

@Jacobolus Yep, I know that all you did was look for a "clean" source for the Taylor expansion of the CDF and choose what, based on the abstract, seemed like the perfect one; I was just playing Devil's advocate in favor of Brown & Brown and replying to Bert Niehaus implicit claim that their formula is "incorrect" ("If that is correct, the cumulative distribution function would have the property ${\displaystyle F(0)=0}$") / to your suggestion that it is "wrong" ("their getting the constant wrong is inexplicable and confusing"). To me that source is terribly worded, but not necessarily wrong. However, I do think that it being terribly worded is a very good reason not to use it as a source (which is a shame though, since it would have been perfect). Malparti (talk) 17:24, 24 February 2026 (UTC)

By the way, if you sign up for WP:LIBRARY you can get the full pdf of this paper here. It's frankly not a very good paper, with a giant mess of jumbled scratch work twiddling various sums, and not much explanation of the context or purpose. It's unfortunately often difficult to find published papers saying obvious and trivial things. –jacobolus (t) 17:40, 24 February 2026 (UTC)

Thanks for the tip! :) Malparti (talk) 18:42, 24 February 2026 (UTC)

I hunted up a different source that mentions this (again, in passing). –jacobolus (t) 19:37, 28 February 2026 (UTC)

Taylor series, existence and uniqueness

Existence and uniqueness is a question for representations - in our context for ${\displaystyle f(z)=\sum _{k=0}^{\infty }{\frac {f^{(k)}(z_{0})}{n!}}(z-z_{0})^{k}}$. Both are answered with complex analysis:

• (Existence) Cauchy's integral formula
• (Uniqueness) identity theorem

Would recommend to mention both. Giving Cauchy credits for his ideas makes sense to me. Furthermore the existence of the Taylor series fails in general on ${\displaystyle \mathbb {R} }$. A counterexample for the failure in ${\displaystyle \mathbb {R} }$ is:

$${\displaystyle f(z):=|z|\cdot z^{n}{\mbox{ with }}z\in \mathbb {R} }$$

${\displaystyle f}$ is ${\displaystyle n}$ times differentiable in ${\displaystyle \mathbb {R} }$ but not ${\displaystyle n+1}$ times, because

$${\displaystyle f^{(n)}(z)=(n+1)!\cdot |z|.}$$

${\displaystyle f^{(n)}}$ is not differentiable in 0 in ${\displaystyle \mathbb {R} }$ and nowhere complex differentiable on ${\displaystyle \mathbb {C} }$.

Recommendation to community: Would recommend to mention the existence and uniqueness for the Taylor series in the context of ${\displaystyle \mathbb {C} }$ and give Cauchy proper credits for his ideas. Bert Niehaus (talk) 07:26, 2 March 2026 (UTC)

@Bert Niehaus Can you please find a source specifically about the normal distribution discussing these topics? Otherwise, if literally every one of the hundreds (thousands?) of sources about this didn't think it was worth mentioning in that context, maybe we shouldn't be the first. –jacobolus (t) 08:30, 2 March 2026 (UTC)

IMHO, Taylor series existence, uniqueness, and entirety for exp(x) or x² / 2 is well-established and not a subject for this article. Taylor series existence, uniqueness, and entirety for the composition of two entire functions (such as these two functions) is also well-established — for example see the first paragraph of the the lede of the article for entire functions — and not a subject for this article. Likewise, termwise integration of an entire function is a well-established process with well-known ramifications, such as being unique up to the constant of integration, and is also not a subject for this article.

Personally, I think that even mentioning that these are well known results is overkill for this article. If enough editors disagree, perhaps we could add a clause to some sentence to indicate that the existence, uniqueness, and entirety of the Taylor series of the normal function cumulative probability distribution follows from standard techniques in complex analysis applied to entire functions. But, really IMHO, devoting an entire sentence to this effort would be overkill. —Quantling (talk | contribs) 16:18, 2 March 2026 (UTC)