X1,X2,X3,X4 should have independent distributions. They should be between 0 and 1 and all add up to 1. Is this still possible with Robert's method?
Thanks On Wed, Mar 26, 2008 at 12:52 PM, Ted Harding <[EMAIL PROTECTED]> wrote: > On 26-Mar-08 20:13:50, Robert A LaBudde wrote: > > At 01:13 PM 3/26/2008, Ala' Jaouni wrote: > >>I am trying to generate a set of random numbers that fulfill > >>the following constraints: > >> > >>X1 + X2 + X3 + X4 = 1 > >> > >>aX1 + bX2 + cX3 + dX4 = n > >> > >>where a, b, c, d, and n are known. > >> > >>Any function to do this? > > > > 1. Generate random variates for X1, X2, based upon whatever > > unspecified distribution you wish. > > > > 2. Solve the two equations for X3 and X4. > > The trouble is that the original problem is not well > specified. Your suggestion, Robert, gives a solution > to one version of the problem -- enabling Ala' Jaouni > to say "I have generated 4 random numbers X1,X2,X3,X4 > such that X1 and X2 have specified distributions, > and X1,X2,X3,X4 satisfy the two equations ... ". > > However, suppose the real problem was: let X2,X2,X3,X4 > have independent distributions F1,F2,F3,F4. Now sample > X1,X2,X3,X4 conditional on the two equations (i.e. from > the coditional density). That is a different problem. > > As a slightly simpler example, suppose we have just X1,X2,X3 > and they are independently uniform on (0,1). Now sample > from the conditional distribution, conditional on > X1 + X2 + X3 = 1. > > The result is a random point uniformly distributed on the > planar triangle whose vertices are at (1,0,0),(0,1,0),(0,0,1). > > Then none of X1,X2,X3 is uniformly distributed (in fact > the marginal density of each is 2*(1-x)). > > However, your solution would work from either point of > view if the distributions were Normal. > > If X1,X2,X3,X4 were neither Normally nor uniformly > distributed, then finding or simulating the conditional > distribution would in general be difficult. > > Ala' Jaouni needs to tell us whether what he precisely > wants is as you stated the problem, Robert, or whether > he wants a conditional distribution for given distributions > if X1,X2,X3,X4, or whether he wants something else. > > Best wishes to all, > Ted. > > -------------------------------------------------------------------- > E-Mail: (Ted Harding) <[EMAIL PROTECTED]> > Fax-to-email: +44 (0)870 094 0861 > Date: 26-Mar-08 Time: 19:52:16 > ------------------------------ XFMail ------------------------------ > ______________________________________________ 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.
