On 06-Mar-07 Alberto Monteiro wrote: > Ted Harding wrote: >> >> And, specifically (to take just 2 RVs X and Y), while U = X/(X+Y) >> and V = Y/(A+Y) are two RVs which summ to 1, the distribution of U >> is not the same as the distribution of X conditional on (X+Y = 1). >> > This question
Which question? There are (implicitly) two questions there! > appeared in October 2006, and the answer To the second question (X conditional on X+Y=1) > was the Dirichlet distribution with parameters (1,1,1...1): > > http://en.wikipedia.org/wiki/Dirichlet_distribution > > It's the distribution of uniform U1, U2, ... Un with the > restriction that U1 + U2 + ... + Un = 1. Indeed, and the resulting (U1,U2,...,Un) is uniformly distributed on the simplex U1+U2+...+Un=1. For n>2, however, the resulting marginal distribution of (say) U1 conditional on (U1+U2+...+Un=1) is no longer uniform (that only holds for n=2, as in my example). For n=3 this is easy to see: P[U1 > u1] is the area of the triangular simplex between its vertex at (1,0,0) and the line from (u1,1-u1,0) to (u1,0,0), and this is equal to (1 - u1)^2, so the density of U1 is f(u1) = 2*(1-u1). In general, the marginal density of U1 in the n-dimensional Dirichlet is (n-1)*(1-u1)^(n-2). But the aim was to illustrate Petr Klasterecky's point that "sum(x) is a random variable as well and dividing by sum(x) does not preserve the original distribution data were generated from." namely to show two ways of generating RVs distributed on U1 + U2 + ... + Un = 1, starting from independent RVs, which result on two different distributions, and to give an example where dividing by sum(x) can be seen to "not preserve" the distribution. Indeed, I think there is sometimes a confusion between this question and the really unrelated question: Given non-negative numbers V1, V2, ..., Vn, how can we convert then to a probability distribution? To which the answer is, of course, divide by their sum. With best wishes, Ted. -------------------------------------------------------------------- E-Mail: (Ted Harding) <[EMAIL PROTECTED]> Fax-to-email: +44 (0)870 094 0861 Date: 06-Mar-07 Time: 22:01:34 ------------------------------ XFMail ------------------------------ ______________________________________________ R-help@stat.math.ethz.ch 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.
