[R] generate random numbers subject to constraints

[Ala' Jaouni]

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
<Ted.Harding at manchester.ac.uk> 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.
>
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>  Date: 26-Mar-08                                       Time: 19:52:16
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