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Why are there no examples???? — Preceding unsigned comment added by 2003:E5:1F0C:364F:1419:3A91:707D:104B ( talk) 07:57, 29 May 2023 (UTC)
Oops, the Mar 22 edit by 68.163.167.174 was me (forgot to log in). The original version of this page had no explanation and simply gave two equations (which were the definition of the expected value--someone copied the wrong equations from the reference). I added some explanation and an example. The theorem is simple so that should be enough for people to understand it, though including the proof of it may be a nice addition. digfarenough ( talk) 23:38, 22 March 2009 (UTC)
Anyone know why it is called this? Btyner ( talk) 19:28, 4 April 2009 (UTC)
(outdent) "Further, they noted, this neglects issues of measurability, continuity, and boundedness." How does it "neglect"? That is the point of the statistician's unconsciousness, I believe and the acronym LOTUS alludes to the same unconsciousness, or is it a transcendental state of bliss. The statistician doesn't need to know the math because the mathematical stumbling blocks do not appear on the statistician's beaten track, only on the way of the mathematician.
A proud and theoretically inclined professional statistician might bowdlerize this the Law of the Unconscious Student (LOTUS), because the mere student statistician rests assured (by hers professor?) that no mathematical stumbling block appears in this course, while the pro must navigate those stumbling blocks in daily life ;-)
Vaguely I feel there should be some interchange of the order of two expectations involved, or it it interchange of some expectations and some limit? Until reading this article (November version) today, I would have said that LOTUS may be the law of iterated expectations? [I might be able to explain by reference to some lecture notes where LOTUS is ubiquitous, my notes as a student, which are not at hand. That would not be for wikipedia mainspace, however, as its referent is unpublished.)
Law means many things to many people. If that's a problem then call it the Lesson or the License in your next course of lectures! -- P64 ( talk) 23:07, 13 March 2010 (UTC)
The only place I've seen this name is A First Course in Probability by Sheldon Ross. In my old edition (3rd), in the second exercise under "Theoretical Exercises" in the chapter on "Expectation" (chapter 7), he states:
He goes on to ask the reader to show that a similar formula he calls the "S-pectation" of (replacing by its square in the first integral formula above) is logically inconsistent because it leads to a contradiction (for those who want to try this, he suggests letting X be uniform(0,1) and trying to compute in two different ways). - dcljr ( talk) 03:00, 7 August 2009 (UTC)
I have the 2nd edition, and although Ross is generally a good author, he does not give an adequate description of expectation. Expectation is a calculation of moment; everything about it follows in some way from that simple statement, including the Law of the Unconscious Statistician. When I teach probability to engineers, I show them that many of the definitions/theorems/methods have parallels in mechanics, particularly in applications of moment ... soon after that, they begin getting passing grades. Pf416 ( talk) 17:42, 15 December 2015 (UTC)
I don't understand why we would want to have this page. Can someone explain. 018 ( talk) 13:14, 21 January 2010 (UTC)
In the section "From the perspective of measure" it is written: "This can be demonstrated by starting with simple functions, and extending the result first to bounded measurable functions, then to arbitrary nonnegative measurable functions, and finally to arbitrary measurable functions, following the usual steps for for Lesbesgue integration". That is true, but why is it here? This is the sketch of the proof of "change of variables formula" and its right place is rather in " pushforward measure#Main property: Change of variables formula", isn't it? Boris Tsirelson ( talk) 13:20, 22 March 2013 (UTC)
LOTUS is a tremendously foundational tool in expectations, and it comes up in essentially every book and introductory course that introduces the idea of expectation. One major detail about this theorem, which is mentioned in the article and which informs its name, is that it is often treated as a definition rather than a proven theorem. However, there are straightforward proofs, at least of very slightly special cases. So, after thinking for a long while and consulting Wikipedia:WikiProject Mathematics/Proofs at length, I decided that it would be very helpful in this page to include a short informal sketch of how LOTUS is proven. Therefore, I am about to add exactly that. - Astrophobe ( talk) 07:29, 6 November 2018 (UTC)
What sources are there for the name besides a pdf document by Bengt Ringner that's never been published, except insofar as putting it on the author's web site constitutes a sort of publication? Michael Hardy ( talk) 20:20, 25 November 2018 (UTC)
For the continuous case, the input x in g(x) can be mapped to a linear 0 to 1 range if you solve the CDF FX(x) for x and plug x into g(x) to get g(FX). CDF's are always uniform in their 0 to 1 range. I think this is sufficient to directly write down the expected value, then use chain rule to prove the standard form.
This is a lot easier to solve than the standard form for my application (feedback control of the expected value which usually needs to use FX as part of g(x) to correctly estimate the error from the target value in each sample of x). It seems equally easy as the standard form for other applications. You can use u substitution on the standard form to derive this equation. Ywaz ( talk) 15:21, 1 June 2022 (UTC)
This article is rated Start-class on Wikipedia's
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Why are there no examples???? — Preceding unsigned comment added by 2003:E5:1F0C:364F:1419:3A91:707D:104B ( talk) 07:57, 29 May 2023 (UTC)
Oops, the Mar 22 edit by 68.163.167.174 was me (forgot to log in). The original version of this page had no explanation and simply gave two equations (which were the definition of the expected value--someone copied the wrong equations from the reference). I added some explanation and an example. The theorem is simple so that should be enough for people to understand it, though including the proof of it may be a nice addition. digfarenough ( talk) 23:38, 22 March 2009 (UTC)
Anyone know why it is called this? Btyner ( talk) 19:28, 4 April 2009 (UTC)
(outdent) "Further, they noted, this neglects issues of measurability, continuity, and boundedness." How does it "neglect"? That is the point of the statistician's unconsciousness, I believe and the acronym LOTUS alludes to the same unconsciousness, or is it a transcendental state of bliss. The statistician doesn't need to know the math because the mathematical stumbling blocks do not appear on the statistician's beaten track, only on the way of the mathematician.
A proud and theoretically inclined professional statistician might bowdlerize this the Law of the Unconscious Student (LOTUS), because the mere student statistician rests assured (by hers professor?) that no mathematical stumbling block appears in this course, while the pro must navigate those stumbling blocks in daily life ;-)
Vaguely I feel there should be some interchange of the order of two expectations involved, or it it interchange of some expectations and some limit? Until reading this article (November version) today, I would have said that LOTUS may be the law of iterated expectations? [I might be able to explain by reference to some lecture notes where LOTUS is ubiquitous, my notes as a student, which are not at hand. That would not be for wikipedia mainspace, however, as its referent is unpublished.)
Law means many things to many people. If that's a problem then call it the Lesson or the License in your next course of lectures! -- P64 ( talk) 23:07, 13 March 2010 (UTC)
The only place I've seen this name is A First Course in Probability by Sheldon Ross. In my old edition (3rd), in the second exercise under "Theoretical Exercises" in the chapter on "Expectation" (chapter 7), he states:
He goes on to ask the reader to show that a similar formula he calls the "S-pectation" of (replacing by its square in the first integral formula above) is logically inconsistent because it leads to a contradiction (for those who want to try this, he suggests letting X be uniform(0,1) and trying to compute in two different ways). - dcljr ( talk) 03:00, 7 August 2009 (UTC)
I have the 2nd edition, and although Ross is generally a good author, he does not give an adequate description of expectation. Expectation is a calculation of moment; everything about it follows in some way from that simple statement, including the Law of the Unconscious Statistician. When I teach probability to engineers, I show them that many of the definitions/theorems/methods have parallels in mechanics, particularly in applications of moment ... soon after that, they begin getting passing grades. Pf416 ( talk) 17:42, 15 December 2015 (UTC)
I don't understand why we would want to have this page. Can someone explain. 018 ( talk) 13:14, 21 January 2010 (UTC)
In the section "From the perspective of measure" it is written: "This can be demonstrated by starting with simple functions, and extending the result first to bounded measurable functions, then to arbitrary nonnegative measurable functions, and finally to arbitrary measurable functions, following the usual steps for for Lesbesgue integration". That is true, but why is it here? This is the sketch of the proof of "change of variables formula" and its right place is rather in " pushforward measure#Main property: Change of variables formula", isn't it? Boris Tsirelson ( talk) 13:20, 22 March 2013 (UTC)
LOTUS is a tremendously foundational tool in expectations, and it comes up in essentially every book and introductory course that introduces the idea of expectation. One major detail about this theorem, which is mentioned in the article and which informs its name, is that it is often treated as a definition rather than a proven theorem. However, there are straightforward proofs, at least of very slightly special cases. So, after thinking for a long while and consulting Wikipedia:WikiProject Mathematics/Proofs at length, I decided that it would be very helpful in this page to include a short informal sketch of how LOTUS is proven. Therefore, I am about to add exactly that. - Astrophobe ( talk) 07:29, 6 November 2018 (UTC)
What sources are there for the name besides a pdf document by Bengt Ringner that's never been published, except insofar as putting it on the author's web site constitutes a sort of publication? Michael Hardy ( talk) 20:20, 25 November 2018 (UTC)
For the continuous case, the input x in g(x) can be mapped to a linear 0 to 1 range if you solve the CDF FX(x) for x and plug x into g(x) to get g(FX). CDF's are always uniform in their 0 to 1 range. I think this is sufficient to directly write down the expected value, then use chain rule to prove the standard form.
This is a lot easier to solve than the standard form for my application (feedback control of the expected value which usually needs to use FX as part of g(x) to correctly estimate the error from the target value in each sample of x). It seems equally easy as the standard form for other applications. You can use u substitution on the standard form to derive this equation. Ywaz ( talk) 15:21, 1 June 2022 (UTC)