tag:blogger.com,1999:blog-3891434218564545511.post1136581696892525716..comments2024-03-27T20:37:09.185-05:00Comments on Alexander Pruss's Blog: Uniform distributionsAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-3891434218564545511.post-3878680654122965742013-06-07T11:18:42.906-05:002013-06-07T11:18:42.906-05:00I know nothing about probability, and I can’t foll...I know nothing about probability, and I can’t follow your whole post, but here is a probably worthless idea. <br /><br />We are confronted with two random variables X and Y, and we wonder how to think about them probabilistically. We do it this way: take a random sample of N values of X, where N is some finite number. The exact size of N doesn’t matter. Call the (non-continuous) variable describing the sample X*. Do the same for Y and Y*. I think it is now true to say that P(X*=0.1)=P(X*=0.2) while P(Y*=0.2)>P(Y*=0.1).<br /><br />Then we say that for continuous variables, the probability properties are the properties of finite random samples of the variables. <br /><br />This accords with what we intuitively want to say about the variables. I don’t know what all it would mess up.<br />Heath Whitehttps://www.blogger.com/profile/13535886546816778688noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-3618366559556386722013-06-06T18:54:33.394-05:002013-06-06T18:54:33.394-05:00Regarding the final comment, in fact it's poss...Regarding the final comment, in fact it's possible to get a random variable whose distribution is the same as Y's but whose pointwise infinitesimal (and I expect conditional--but I haven't worked that out) probabilities are plausibly more like we'd expect from the graph.<br /><br />Let X be uniform and let X' be an independent copy of X. Let α = P(X=a) for all a, where α is infinitesimal. Let Y' = max(X,X'). Then it's pretty easy to check that Y' has the same probability distribution as Y. But pointwise, Y' is rather different from Y, when Y is as in the post. For it's easy to check that if we allow these infinitesimal calculations, then P(Y'=a) = 2aα + O(α^2), so to a first order approximation in α, we have exactly what the graph would lead us to think. On the other hand, P(Y=a) = α for all a.<br /><br />I wonder if the pointwise data carries some additional information about a random process beyond what is carried by the distributional data. Or if the pointwise stuff is all nonsense. My previous work on infinitesimals suggests the pointwise stuff is junk, but perhaps the conditional probability version works better.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.com