Talk:Moment-generating function

message: Spelling question:  I've never (before now) seen the name spelled with a hyphen.  Searches of Math Reviews (MathSciNet) and Current Index to Statistics show an overwhelming preference for no hyphen.  Should the title, at least, be changed (move the article to "Moment generating function" with a redirect)? Zaslav 18:08, 8 December 2006 (UTC)
replies: text: Sir Ronald Fisher always used the hyphen in "moment-generating function".  This is an instance of the fact that in this era the traditional hyphenation rules are usually not followed in scholarly writing, nor in advertising or package labelling, although they're still generally followed in newspapers, magazines, and novels.  This particular term seldom appears in novels, advertisements, etc.  Personally I prefer the traditional rules because in some cases they are a very efficient disambiguating tool. Michael Hardy 20:03, 8 December 2006 (UTC)
replies: text:  Some literature sources also seem to use the hyphen. Example Paul L. Meyer's Introductory Probability and Statistical Applications. Apoorv020 (talk) 20:23, 10 October 2009 (UTC)
replies: text:  We really should have consistency on this throughout the article. I personally never use a hyphen, and my opinion on this is that we should probably just go with the more common usage. Perhaps this is something for RfC? Either way, we could simply state at the beginning of the article that either usage is used in literature. Blahb31 (talk) 22:42, 28 March 2015 (UTC)
thread_title: Hyphenation

message: We should mention the case when X is a vector of random variables or a stochastic process. Jackzhp 22:29, 3 September 2006 (UTC)
replies: text: I've added a brief statement about this.  (Doubtless more could be said about it.) Michael Hardy 03:48, 4 September 2006 (UTC)
thread_title:  Vector of random variables or stochastic process 

message: something should be added about the discrete form of the mgf, no?
M_X(s)=E(e^{sX})=\sum_{k=1}^\infty{p(k)e^{sk}} 24.136.121.150 08:37, 20 January 2007 (UTC)
thread_title:  Discrete form of mgf 

message: It would seem like a good and self-evidently obvious thing to include a link to a wikipedia page which tabulates common moment generating functions (ie: the moment generating functions for common statistical distributions), placing them online.  The information is already there on wikipedia, it would just be a case of organising it a little better.  
Also, there is probably some efficient way in which the set of all possible functions which commonly occur when dealing with statistical distributions can be organised to highly the possibility of inter-relationships (perhaps some mgf's are nested mgfs so that the fact that
 Mgf_3(s)=Mgf_2(Mgf_1(s)) 
could be highlighted in a list of mgf interdependencies...).  
ConcernedScientist (talk) 00:47, 18 February 2009 (UTC)
thread_title: Common Moment Generating Functions

message: We have a theorem that if two mgf's coincide in some region around 0, then corresponding random variables have same distribution. There is however a concern that this statement being true from the point of view of mathematician, is not so reliable from the point of view of applied statistician. McCullagh (1954)McCullagh, P. (1954). Does the moment generating function characterize a distribution? American Statistics, 48, p.208 gives following example:
: f_1(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}, \quad f_2(x) = f_1(x)\Big(1 + \frac{1}{2}\sin(2\pi x)\Big)
with cumulant generating functions
: K_1(t) = t^2/2, \quad K_2(t) = K_1(t) + \ln\Big(1 + \frac{1}{2}e^{-2\pi^2}\sin(2\pi t)\Big)
Although the densities are visibly different, their corresponding cgfs are virtually indistinguishable, with maximum difference less than 1.34•10-9 over the entire range. Thus from numerical standpoint mgfs fail to uniquely determine distribution.
On the other hand Waller (1995)Waller, L.A. (1995). Does the characteristic function numerically distinguish distributions? American Statistics, 49, pp. 150-151 shows that characteristic function does much better job in determining the distribution.
Even from a mathematician's point of view, don't you need the MGF to be infinitely differentiable at 0 for uniqueness? Paulginz (talk) 14:45, 24 November 2009 (UTC)
thread_title:  Uniqueness 

message: The MGF is calculating the raw moments as opposed to the central moments. It already says "moments around the origin" on the page, but perhaps this should be noted explicitly at the end of the introduction ("The moment generating function is named as such, because of its intimate link to the raw moments") and again in the definition (only inserting raw), f.ex. : ...where mn is the nth raw moment. 
Currently the first definition on the moment page is of central moments, so a bit of confusion could occur.
/Jens  — Preceding unsigned comment added by 203.110.235.1 (talk) 23:55, 21 January 2013 (UTC)
thread_title:  raw moments vs. central moments. 

message: Under "definitions" further up on this page, there is some concern that the article is not transparent to someone trying to learn about moment generating functions.  Michael Hardy writes, "It is written for people who already know what probability distributions are and the standard basic facts about them."  While this is true, I think it jumps somewhat between requiring only this basic amount of knowledge and requiring more substantial knowledge.  I would imagine many of my students in my intro probability course would find this article impenetrable at enough places that they would give up trying to learn something from it.  I would think it would be better to put anything that requires knowledge from courses at a comparable level of difficulty or more advanced than an intro probability course at the end. 
:
Specifically, I think it would improve the article to:
# Move the definition of multivariate moment generating functions and the corresponding vector notation.  Also move the vector-valued case under "calculation" to be next to it.
# Don't make the computation in the "general case" the first item under computation.  Move anything concerning Riemann-Stieltjes integration further down.
# Move the theorem about equal moment generating functions implying CDFs equal at almost all points down -- the reference given, Grimmett (and Welsh), doesn't even state it in that generality.  Instead, give the restricted version actually given in Grimmett and Welsh and talk about the more general "outside a set of meaure 0" case further down.
:
Does anyone feel that these changes would not be an improvement?Barryriedsmith (talk) 14:23, 6 October 2015 (UTC)
thread_title:  Too much a mix of lay and advanced knowledge 

message: Substituted at 20:07, 1 May 2016 (UTC)
thread_title: Assessment comment

message: Consider
:\Pr[X \ge t] \le \frac{E[X^k]}{t^k} \le \frac{k!}{t^{2k}} M_{X}(t)  \le e\sqrt{k}\frac{k^k}{e^k t^{2k}} M_X(t) \le et\, e^{-t^2} M_X(t),
under "Other Properties". The estimate
:\frac{k!}{t^{2k}} \le et\, e^{-t^2}
is simply false for many values of k and t (k fixed, t large). 178.38.132.48 (talk) 18:00, 4 December 2017 (UTC)
replies: text: I believe the idea is that the second last expression is valid for any k, so you can choose k=t^2 to get the final expression.  I'm somewhat dubious about the case when X is not non-negative and k is not integer. McKay (talk) 03:06, 11 December 2017 (UTC)
thread_title:  Incorrect estimate in the section "Other Properties" 

message: The article claims that the Expectation must exist in order for the MGF to exist. This is not true. The expectation can exists (e.g., Cauchy, or Log-Normal - where it's positive infinite) but not be finite - and then there is no MGF. 
Also, the link in the Cauchy to Indeterminate form is wrong - since for Cauchy the expectation of e^tX is positive infinite. No indeterminate issues here.  — Preceding unsigned comment added by 109.186.33.244 (talk) 14:44, 30 November 2021 (UTC)
thread_title:  Existence issues