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the question will be, random with what distribution?
I assume with a uniform distribution: each point in the final region has the same probability to be chosen. For a single rectangle this is straight forward:
For multiple rectangles that don't overlap, we can say, we choose the rectangle proportionally to it's area and once the rectangle is chosen, pick a point in there.
But if they overlap it's tricky. The easiest way I can think of is to union the new rectangle with the existing area and only consider the rectangles that are not already covered and apply method 2.- Chris December 03, 2017