# Re: [R] generate random numbers subject to constraints

From: Ala' Jaouni <ajaouni_at_gmail.com>
Date: Wed, 26 Mar 2008 14:26:59 -0700

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.
>
> --------------------------------------------------------------------
> E-Mail: (Ted Harding) <Ted.Harding_at_manchester.ac.uk>
> Fax-to-email: +44 (0)870 094 0861
> Date: 26-Mar-08 Time: 19:52:16
> ------------------------------ XFMail ------------------------------
>

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