How to draw 2 random number simultaneously which are smaller than 1 together?

I am using a formula that has 3 different parameters: alpha, beta and 1-alpha-beta. These 3 parts have to have the same average so also the same probability to be high or low.

I want to draw the alpha and beta simultaneously but together they have to be smaller than 1. It should also be possible that for instance alpha = 0.1 and beta is 0.8.

I already tried to first draw alpha and afterwards draw beta with restriction 1-alpha-beta but this resulted into an alpha that is higher than beta in most cases.

I also tried to just draw 2 random numbers and make a if-loop with the restriction that both should be smaller than 1 but also this resulted into not evenly distributed results since the third part, 1-alpha-beta, was on average smaller than alpha and beta.

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You appear to want to generate a pair of random numbers such that they are uniformly distributed, live in the interval [0,1], AND they have a sum that is less than 1. (See below, because you actually appear to have a third constraint.)

The clue is to think about what this means about where the numbers must live. They must live in the triangle in the (x,y) plane with vertices {[0,0], [0,1], [1,0]}. Thus…

fill([0 0 1],[0 1 0],'r')
axis equal

So ANY point inside that triangle corresponds to an (x,y) pair of numbers that satisfies your requirement. A uniform sampling means that any location in that triangle is equally likely to arise.

The simplest way to do so is by a simple rejection scheme. (There are other ways to generate this sampling, but be careful. because there are many ways to do it wrongly too. If you wanted, I could show you how to do the sampling in other ways, but this answer is already long enough.) Just generate more points in the unit square than you need. Then discard any point that lies above the diagonal. Those that remain satisfy your goal. These rejection methods are often a good choice, because they result in a uniform sampling. And here we will reject roughly 50% of the points generated. So if you want 1000 points, then generate a little more than twice as many as you need.




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