First off, interesting comparison between both methods and a very good point about variance, randomness, and reproducibility.

I have a question about the random interval [0,1], especially with regards to the area method. I take that notation to mean 0 and 1 are included in the interval and are valid outputs of randfun.

Wouldn't the points on the axis (only one of them, your choice) be counted double? If you look at the complete circle, then the correct range would be [-1,1], with the -1 and +1 included. If you divide it up into quarters, you would want to count only _one_ axis per quarter (say the +y for the quadrant shown in the post), because the other axis would be included in the other quadrants (say the +x axis with the next quadrant clockwise). And thinking like that would bring up the question whether to include the origin or not..

Anyway, my question is whether my interpretation of the effect of the random interval is correct? If so, how to handle it? Especially the origin? And would the average value method also be affected? (perhaps restrict the interval there to (0,1]? That is, 0 not included?)