Generate pair of random numbers with respect to a sum constraint?

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Hello, all. So I am still gaining experience with Matlab and am currently trying to generate sequences of two numbers, x and y, such that their sum is <= 1. The naive way I immediately thought to do this would be by 1) generating x within the range [0, .5], and 2) keep generating y until it is < x, which would guarantee my condition, but introduce some significant statistical bias.
Is there some standard, statistically robust way to do this, guys? I would be very grateful for any of your input on this matter.

Accepted Answer

Roger Stafford
Roger Stafford on 22 Feb 2018
Edited: Roger Stafford on 22 Feb 2018
(Corrected) Assuming you restrict x and y to non-negative values, the set of x and y values for which x+y<=1 would be a triangular area in the xy plane. You can obtain an area-wise uniform distribution of x and y with the following.
For generating a single pair:
x = 1-sqrt(rand);
y = (1-x)*rand;
To get row vectors with n elements each:
x = 1-sqrt(rand(1,n));
y = (1-x).*rand(1,n);

More Answers (1)

Jeff Miller
Jeff Miller on 21 Feb 2018
It isn't entirely clear what joint distribution you want for (x,y), but here is one possibility:
x = rand; % uniform 0 to 1
y = (1-x)*rand; % uniform 0 to 1-x
  4 Comments
César
César on 26 Mar 2019
Dear @Roger Stafford,
Could you please give also some explanations of how do y = (1-x).*rand(N,1) give the correcrt distribution please?

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