I have a moving platform which moves in orthogonal axis to move a light detector to take account of positional uncertainty. I have a systematic error in space that is well described by a circular normal distribution (actually its not quite circular but the covariance matrix is almost exacty diagonal). I can think of two ways I could determine the amount by which my platform needs to move. I can either look at the probability as a function of radius so that my platform needs to move by 2.15 times the standard deviation to achieve 90%, this was my first though. The motors on the platform however move in orthogonal axes so I also though that I could look at the marginal distributions in which case for each axis the platform needs to be able to move by 1.64 times the standard deviation to achieve 90% coverage. As it happens this isn't really a concern as the range is much greater than required but it got me thinking.
If I had a positional uncertainty distribution that was a circular normal distribution which method should I use to detemine the amount of movement I need to cover 90% of events. If I think just about the motion in a single axis then, thinking radially, I need to move by 2.15 times the standard deviation however thinking 'marginally' then I only need 1.64 times the standard deviation. Which method is correct?