tag:blogger.com,1999:blog-4068183698747623113.post1769504155936217039..comments2023-10-29T10:40:34.638-04:00Comments on A CS Professor's blog: IndependenceClaire Mathieuhttp://www.blogger.com/profile/10957755706440077623noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-4068183698747623113.post-22644056971802571882012-02-13T07:17:19.745-05:002012-02-13T07:17:19.745-05:00Another belated example: if A, B, C, D are 4 indep...Another belated example: if A, B, C, D are 4 independent events, then A union B is independent of C union D. This can be generalized to intersections, complements, more than 4 events...Pascalhttps://www.blogger.com/profile/14201150679841329835noreply@blogger.comtag:blogger.com,1999:blog-4068183698747623113.post-11291281244947492822012-01-20T01:04:08.661-05:002012-01-20T01:04:08.661-05:00I can't think of an example off-hand. But I re...I can't think of an example off-hand. But I remember finding this formulation useful in a real-life situation. It's often the case that expectations are easier to calculate than probability.Arnabhttp://web.mit.edu/abhatt/wwwnoreply@blogger.comtag:blogger.com,1999:blog-4068183698747623113.post-7748854424461070482012-01-19T15:55:41.986-05:002012-01-19T15:55:41.986-05:00Can you give an example where that would be the ri...Can you give an example where that would be the right way to present independence?Claire Mathieuhttps://www.blogger.com/profile/10957755706440077623noreply@blogger.comtag:blogger.com,1999:blog-4068183698747623113.post-17556061773151472462012-01-19T15:53:29.534-05:002012-01-19T15:53:29.534-05:00Here's another potential way. If X_1, ..., X_n...Here's another potential way. If X_1, ..., X_n are bounded, real-valued random variables, then they are independent if and only if E[X_1^m_1 X_2^m_2 ... X_n^m_n] = E[X_1^m_1]*E[X_1^m_2]*...*E[X_n^m_n] for all non-negative integers m_1, ..., m_n. This follows by Weirstrass' approximation theorem.Arnabhttp://web.mit.edu/abhatt/wwwnoreply@blogger.comtag:blogger.com,1999:blog-4068183698747623113.post-76156513853469438802012-01-18T11:04:00.220-05:002012-01-18T11:04:00.220-05:00There are little lemmas that can sometimes be use...There are little lemmas that can sometimes be used towards actual proofs of independence.<br />For instance, if X and Y are independent random variables and f is a function, then the random variables f(X) and f(Y) are independent (really!).Pascalhttps://www.blogger.com/profile/14201150679841329835noreply@blogger.comtag:blogger.com,1999:blog-4068183698747623113.post-91126358789235331892012-01-18T10:55:43.266-05:002012-01-18T10:55:43.266-05:00I like the definition Pr[A|B]=Pr[A] better than Pr...I like the definition Pr[A|B]=Pr[A] better than Pr[A and B]=Pr[A]Pr[B], even though it breaks symmetry. It's the definition I usually use in proofs.<br /><br />My main step towards rigor when such a claim, proved in English, looks slightly dubious, is to get rid of probability: break things down into elementary events, look at all possibilities, and reduce the problem to counting events one by one. Tedious, but necessary (and sufficient) when I am suspicious.Claire Mathieuhttps://www.blogger.com/profile/10957755706440077623noreply@blogger.comtag:blogger.com,1999:blog-4068183698747623113.post-15173393809389367292012-01-18T10:08:38.734-05:002012-01-18T10:08:38.734-05:00Forgive my ignorance, but I always argue (admitted...Forgive my ignorance, but I always argue (admittedly using English) that Pr[A|B] = Pr[A] to establish that A and B are independent. Is there a more rigorous method?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-4068183698747623113.post-58134953406006756692012-01-18T08:58:29.024-05:002012-01-18T08:58:29.024-05:00I had a lot of trouble figuring out to say about i...I had a lot of trouble figuring out to say about independence when teaching elementary probability.<br /><br />Usually, textbooks say, "Events A and B are defined to be independent if Pr[A and B] = Pr[A]Pr[B]".<br /><br />That's fine and all, but 99.9% of the time when independence is used in proofs, the proofs look like this:<br /><br />"Let A be the event ... and let B be the event ... Now A and B are independent because [natural language explanation]. Therefore we can use the fact that Pr[A and B] = Pr[A]Pr[B]..."<br /><br />Obviously, this looks circular and invalid. Yet that's how mathematicians write.<br /><br />So you need to explain that there are techniques for proving A and B are independent without relying on the definition... so that you can deduce the definition.Ryan O'Donnellhttps://www.blogger.com/profile/01760886084136827344noreply@blogger.com