Trying to come up with a distribution D(1/R, R) that meets your requirements isn't the right way to go about this, or at least it's extremely difficult. Let's say you consider the ratio of X-to-Y, with a range [1,R] that satisfies the constraint XY = A (some area). When you adjust the X-to-Y ratio range to be [1,1/R], it's really the same as the ratio of Y-to-X, over the range [1,R]. So the issue here with a single distribution is that you're dealing with what I would see as a local "compression" over [1/R,1], since it's equivalent to [1,R] if you just switch X and Y however you have two different size ranges for something that's geometrically equivalant. This problem is formalized by the constraint you mention in your post: P(L<1/a)=P(L>a). I'm sure that such a distribution exists, however coming up with it probably involves some complex math and statistics.

But using the apparent symmetry around 1 of [1,R] for X-to-Y and [1,R] for Y-to-X, we can decompose the problem into two uniform distributions M~U(-1,1) and L~U(1,R). L will be the final ratio, however if M>=0 then the ratio is for X-to-Y and if M<0 then the ratio is for Y-to-X. Using your constraints, you can come up with a system of equations to solve for either X or Y as a function of the ratio and the area A, and then using this ratio you can solve for the other length. This construction satisfies all your requirements.

[Edited by - Zipster on October 7, 2005 8:10:22 AM]

The requirement that E(L)=1 seems very arbitrary to me. You probably need something else and didn't think it through. Did you put that requirement so you don't get too many or too few oblong rectangles? Any natural distribution with E(L)=1 will generate more values under 1 than over 1.

It looks to me that the natural quantity to consider in this problem is not the ratio of the too lengths, but the angle that a diagonal forms with one of the sides. That angle would be between atan(1/R) and atan(R), but it turns out that those two angles can be written as 45º-delta and 45º+delta. I think a uniform distribution U(45º-delta, 45º+delta) for the angle will do just fine, or any other symmetric distribution around 45º.

Those equations don't take the area constraint into account. Whatever solution you use to find the ratio, you're still going to have to go back and solve for one side and then the other. The angle-based method looks clever, though. As long as you work off symmetry things should be fine.