Re: Random Sampling in 2D with a Known Distribution

From: David Jones (dajxxx_at_ceh.ac.uk)
Date: 10/27/04 

Date: Wed, 27 Oct 2004 13:01:35 +0100

andre wrote:
>
> The "analytical" way to get what you want would be to find two
> multi-variate functions f1(x1,x2,...,xn) and f2(x1,x2,...,xn)
> such that your density P(y1,y2) is given by
>
> P(y1,y2) = integral( dirac( y1-f1(x1,...,xn) )*dirac(
> y2-f2(x1,...,xn) ) )
>
> where the integral is over all variables x1=0..1, x2=0..1, ...
> ,xn=0..1.
> The minimal number of variables you would need is 2. But in general
> it is too difficult to find such functions.
>

 But the most straightforward "analytical" way is to proceed via the
representation of the joint distribution as a single marginal
distribution (for one of the existing variables) and the condition
distribution of the second given the first. Thus you don't have to
"find" anything. However, other approaches such as those suggested,
can well give someting more computationally efficient. If you problem
coincides with one of the well-established bivariate or muultivariate
distrubitions, you may find that simulation procedures have already
been studied.

David Jones