generating random numbers respect to a certain constraint

hello everybody, I want to generate two numbers ( x1 and x2 ) which have a normal distribution which always respect an equality ( x1+x2=0.5 ).

Appreciate your kind comments.

Answer 1

[John D'Errico]

Trivial? Although really, you have not provided sufficient information.

What are the parameters of that distribution along the constraint line? Just saying the distribution is normal is not sufficient. Regardless, normal distributions are so easy to work with that the answer is easy.

V = [1 1];V = V/norm(V);
xy = randn(10000,2);
xy = xy - xy*V'*V + [0.25 0.25];

min(sum(xy,2))
ans =
          0.5

max(sum(xy,2))
ans =
          0.5

As I said, normal distributions are so easy to work with, that this is trivial. The rows of xy represent points along the constraint line, normally distributed. The variance of those points along the constraint line is 1.

Or, I could have done it by creating a singular covariance matrix. You need to be careful there, because the standard methods to create normal random variables can fail, if the covariance matrix is just slightly not positive definite. I have posted a tool called nearestSPD on the file exchange though.

[Vahid_68]

Appreciate in advance for your answer, would you mind please send me the mathematics logic behind this method, or any reference for this method?

[John D'Errico]

Basic linear algebra. Its just a projection into a subspace.

Answer 2

[Walter Roberson]

x1 = randn();
x2 = 0.5 - x1;

[Vahid_68]

Thanks Walter, but the problem is generating random numbers such that they follow a certain constraints, moreover the numbers should have a normal distribution also. each number(set of numbers) should have its own mean and standard deviation.