... To pick up on what Mark has said: it strikes me that this is related to the simplex, where the bounded nature of the vector space means that normal arithmetical operations (i.e. Euclidean) don't work---that is, they can be used, but the results are wrong. Covariances and correlations for instance, are wrong, something that Pearson noted more than a century ago.
Taking logs of ratios of numbers (say a number divdided by geometric mean of the other numbers, as in Aitchison's set of transformations) in this space transfers everything to Euclidean space, so "squaring" the problem. This is why taking logs fixes things ?? Mark. statquant wrote: > > Duncan: I think I see what you're saying but the strange thing is that if > you use the utility function log(x) rather than x, then the expected > values > are equal. Somehow, if you are correct and I think you are, then taking > the > log , "fixes" the distribution of x which is kind of odd to me. I'm sorry > to > belabor this non R related discussion and I won't say anything more about > it > but I worked/talked on this with someone for about a month a few years > ago > and we gave up so it's interesting for me to see this again. > > Mark > > -----Original Message----- > From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] > On > Behalf Of Duncan Murdoch > Sent: Tuesday, August 26, 2008 8:15 AM > To: Jim Lemon > Cc: r-help@r-project.org; Mario > Subject: Re: [R] Two envelopes problem > > On 26/08/2008 7:54 AM, Jim Lemon wrote: >> Hi again, >> Oops, I meant the expected value of the swap is: >> >> 5*0.5 + 20*0.5 = 12.5 >> >> Too late, must get to bed. > > But that is still wrong. You want a conditional expectation, > conditional on the observed value (10 in this case). The answer depends > on the distribution of the amount X, where the envelopes contain X and > 2X. For example, if you knew that X was at most 5, you would know you > had just observed 2X, and switching would be a bad idea. > > The paradox arises because people want to put a nonsensical Unif(0, > infinity) distribution on X. The Wikipedia article points out that it > can also arise in cases where the distribution on X has infinite mean: > a mathematically valid but still nonsensical possibility. > > Duncan Murdoch > > ______________________________________________ > R-help@r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide > http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. > > ______________________________________________ > R-help@r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide > http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. > > -- View this message in context: http://www.nabble.com/Two-envelopes-problem-tp19150357p19163195.html Sent from the R help mailing list archive at Nabble.com. ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
