tag:blogger.com,1999:blog-3758330678390419129.post2239829351442732994..comments2024-08-10T19:58:25.387-04:00Comments on EconomicPolicyJournal.com: This Should Be EasyRobert Wenzelhttp://www.blogger.com/profile/14296920597416905488noreply@blogger.comBlogger87125tag:blogger.com,1999:blog-3758330678390419129.post-20118242355870513592012-04-09T10:45:12.475-04:002012-04-09T10:45:12.475-04:00ziragt, you're welcome. I guess this just goe...ziragt, you're welcome. I guess this just goes to show that even when both sides are confident in their positions, this sort of discussion can still be fruitful for them both.<br /><br />Given your source, I will gladly accept that the measure-theoretical definition you've given is widely used--though if you could check the actual text, that would be doubly reassuring. I had hoped to David Gnoreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-83753567899250180992012-04-08T20:42:31.831-04:002012-04-08T20:42:31.831-04:00David, I am not coming up with an ad hoc definitio...David, I am not coming up with an ad hoc definition. I am trying to state that P(X=x) is defined, using completely standard concepts. I am not sure what else there is to say about this.<br /><br />As for a reference, I suppose that Kai-Lai Chung's textbook, "A Course in Probability Theory", would state it, although I don't have my copy on hand to verify this. Most graduate ziragthttp://mises.org/Community/blogs/zirag/default.aspxnoreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-36975252508159104962012-04-08T13:20:03.483-04:002012-04-08T13:20:03.483-04:00ziragt, I'm afraid you're somewhat mistake...ziragt, I'm afraid you're somewhat mistaken about what I've written. As I wrote to Bob, I am not saying that it is impossible to come up with some definition of P(X=x) for continuous distributions, but merely that it doesn't have a preexisting definition. Nor am I disputing that if it were to be defined, that it would be 0--in fact, the definition that I gave to Bob as a David Gnoreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-34049888121036724252012-04-08T11:33:50.886-04:002012-04-08T11:33:50.886-04:00David G, you are simply wrong. For one, Bob's ...David G, you are simply wrong. For one, Bob's example cumulative distribution only applies to continuous distributions, for obvious reasons. Saying that it fails because it does not apply to discrete distributions is nonsense. He is giving a special case where the result is true because you refuse to take the definition given on Wikipedia and in all probability textbooks I have ever seen ziragthttp://mises.org/Community/blogs/zirag/default.aspxnoreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-18138173049454103352012-04-07T13:40:23.272-04:002012-04-07T13:40:23.272-04:00Bob, I'm afraid your attempted definition does...Bob, I'm afraid your attempted definition doesn't quite work. The problem is that it produces P(X=x)=0 for discrete distributions as well. A better definition would be the limit as h goes to zero of F(x+h)-F(x-h), where F is the cumulative distribution function. This, however, runs into the problem of boundary points, so we could define a one-sided limit for those cases. However, my David Gnoreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-38639868294186000542012-04-07T11:45:43.117-04:002012-04-07T11:45:43.117-04:00David Gordon wrote: If it is of any interest, I am...David Gordon wrote: <i>If it is of any interest, I am not the "David G." who has posted above.</i><br /><br />I agree that there is a 0% probability that the careful academic I know would, with such confidence, utter demonstrably false things. However, I won't go so far as to say it's literally impossible that you two are the same guy.Bob Murphyhttps://www.blogger.com/profile/04001108408649311528noreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-85928709494795761382012-04-07T08:34:52.490-04:002012-04-07T08:34:52.490-04:00"that is incorrect: it is undefined."
I..."that is incorrect: it is undefined."<br /><br />I agree with that but your problem statement did not say <br />"...for any particular number in that range, there is an UNDEFINED chance you will hit it." Your problem statement said 0%.<br /><br />You also refer to a nifty proof posted on your blog, not disagreeing with it at all - but doesn't it prove by contradiction thatliquifactionnoreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-28359519691442592522012-04-07T06:49:28.231-04:002012-04-07T06:49:28.231-04:00OK, David G, you said that any math PhD would agre...OK, David G, you said that any math PhD would agree with you. Now, after being pointed to several who don't, you completely change tack and ask me for a definition that yields this result. David, I am NOT a PhD mathematician. In a field in which I am not an expert, I take the word of experts. Apparently, you found it discourteous for me to ask where your PhD is from, and what your full name gcallahhttps://www.blogger.com/profile/10065877215969589482noreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-82904005304365155232012-04-07T04:42:33.481-04:002012-04-07T04:42:33.481-04:00Sorry for the late reply, I did not realize this p...Sorry for the late reply, I did not realize this post had generated such controversy.<br /><br />Im not discusing the math concepts used so your response makes no sense (again). What Im saying is that you are playing with the concepts of absolute cero and cero as a limit to give the impression of writting something paradoxical and very meaningful when your comment is neither. To someone who Unknownhttps://www.blogger.com/profile/02667553727717904159noreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-14428525286858130692012-04-06T23:52:16.176-04:002012-04-06T23:52:16.176-04:00If it is of any interest, I am not the "David...If it is of any interest, I am not the "David G." who has posted above.David Gordonhttps://www.blogger.com/profile/03491884384604322669noreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-28375448530646713132012-04-06T23:22:00.296-04:002012-04-06T23:22:00.296-04:00Oops David G. I can't see my comment now (stil...Oops David G. I can't see my comment now (still awaiting moderation), but I think I skipped a step: You *define* the probability that X will fall in between a and b, by integrating the probability density function over x, and evaluating at those end points. So the solution to that definition gives you the formula of (b-a).Bob Murphyhttps://www.blogger.com/profile/04001108408649311528noreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-87496475991287121892012-04-06T23:11:28.036-04:002012-04-06T23:11:28.036-04:00David G. wrote: Gene, if you are so sure that my s...David G. wrote: <i>Gene, if you are so sure that my statement was false and have access to great experts in the field of probability, then by all means provide a nontrivial definition of P(X=x) for continuous distributions (by nontrivial, I mean that you don't simply declare P(X=x)=0, but that you can derive that conclusion from the definition). </i><br /><br />OK it's late and I am not aBob Murphyhttps://www.blogger.com/profile/04001108408649311528noreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-70486834053894792932012-04-06T20:24:10.308-04:002012-04-06T20:24:10.308-04:00What's amazing to me is that with all the incr...What's amazing to me is that with all the incredibly important posts about monetary policy, Fed abuses, civil rights abuses, etc. on this blog, the blog entry that gets the record number of replies is some asinine argument over infinitesimals (literally). In the end, you may come up with some answer that makes equations work, but nobody can tell you what it really means -- like "photonsAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-57468230311597273152012-04-06T19:50:49.464-04:002012-04-06T19:50:49.464-04:00Gene, the wikipedia article you cited contains a f...Gene, the wikipedia article you cited contains a further passage:<br /><br />"This apparent paradox is resolved by the fact that the probability that X attains some value within an infinite set, such as an interval, cannot be found by naively adding the probabilities for individual values. <b>Formally, each value has an infinitesimally small probability, which statistically is equivalent to Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-34804075203372839362012-04-06T19:27:34.174-04:002012-04-06T19:27:34.174-04:00Well, one obvious proof is that the integral from ...Well, one obvious proof is that the integral from x to x is 0 in either the Lebesgue or Riemann integral. Getting deeper into this problem requires understanding how probability spaces are defined and why, which requires a bit more thought.<br />It's also strange to declare that something is not well-defined when one does not like the definition.ziragtnoreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-64499151974239566912012-04-06T16:53:10.936-04:002012-04-06T16:53:10.936-04:00Bob, my point is that Gene's statement is mean...Bob, my point is that Gene's statement is meaningless. In the same way, "gobbledegook fluoresces" is meaningless rather than false because "gobbledegook" is a meaningless word.<br /><br />Yes, I am saying that P(X=0) is not well-defined for a continuous distribution and so is not 0.<br /><br />Gene, if you are so sure that my statement was false and have access to great David Gnoreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-61111308540434958902012-04-06T06:38:38.212-04:002012-04-06T06:38:38.212-04:00That would not be a very interesting post Gene.That would not be a very interesting post Gene.liquifactionnoreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-20770200143637177822012-04-06T05:35:04.471-04:002012-04-06T05:35:04.471-04:00"Due to the lack of precision in the measurem..."Due to the lack of precision in the measurement (i.e. only one digit)..."<br /><br />Again, confusing math with experimental science.gcallahhttps://www.blogger.com/profile/10065877215969589482noreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-67441692840557037972012-04-06T05:34:00.367-04:002012-04-06T05:34:00.367-04:00JTG, you have confused experimental science with m...JTG, you have confused experimental science with mathematics: the concept of "decimal places of precision" only applies in experimental science.gcallahhttps://www.blogger.com/profile/10065877215969589482noreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-79474733310275397482012-04-06T05:32:39.899-04:002012-04-06T05:32:39.899-04:00"I understand that any finite number divided ..."I understand that any finite number divided by infinity is taken as 0.."<br /><br />That is incorrect: it is undefined.<br /><br />"It does not exist because it has an infinitely small chance of existing..."<br /><br />No, this is all wrong. See the Wikipedia article on continuous distribution probability.gcallahhttps://www.blogger.com/profile/10065877215969589482noreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-20445437601125252562012-04-06T05:30:25.022-04:002012-04-06T05:30:25.022-04:00Thanks for asking, Richie! I am posting today from...Thanks for asking, Richie! I am posting today from the British POlitical Studies Association conference in Belfast, where I presented my paper on Bishop Berkeley and have been pitching my fourth book. But after this, it's back to teaching at Purchase College, editing, and getting cracking on that fourth book!gcallahhttps://www.blogger.com/profile/10065877215969589482noreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-33829701567455577942012-04-06T05:28:01.734-04:002012-04-06T05:28:01.734-04:00@David G: "Actually, Gene, if you ask any mat...@David G: "Actually, Gene, if you ask any math Ph.D., they will tell you that for continuous distributions P(X=x) isn't defined."<br /><br />That's funny, David G, because I GOT the dart example from my boss, a math PhD and the author of two books on probability theory, Randolph Nelson. Clay Shonkwiller, a math PhD, came to my original post and called this a good example. I was gcallahhttps://www.blogger.com/profile/10065877215969589482noreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-23318160084242616692012-04-06T01:29:10.279-04:002012-04-06T01:29:10.279-04:00Richie asked, "Does Callahan have a job, or d...Richie asked, <i>"Does Callahan have a job, or does he sit in a library all day long?"</i><br /><br />What if Callahan is a librarian in a wheelchair?Bob Murphyhttps://www.blogger.com/profile/04001108408649311528noreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-28586884904090965572012-04-05T23:41:32.548-04:002012-04-05T23:41:32.548-04:00David G. are you sure about that? For the non-impo...David G. are you sure about that? For the non-important point: I think what you mean to say is that Gene's statement was false, not that it was meaningless. If I claim, "The largest number in the range [0, 1) is 1," I didn't say something meaningless, I said something false. (Note that in the example, I am trying to assign a value to an undefined concept, namely the largest Bob Murphyhttps://www.blogger.com/profile/04001108408649311528noreply@blogger.comtag:blogger.com,1999:blog-3758330678390419129.post-27872137081127309642012-04-05T19:28:19.055-04:002012-04-05T19:28:19.055-04:00Does Callahan have a job, or does he sit in a libr...Does Callahan have a job, or does he sit in a library all day long?Richiehttps://www.blogger.com/profile/08214872228234094199noreply@blogger.com