Bounds are found for the distribution function of the sum of squares X2+Y2 where X and
Y are arbitrary continuous random variables. The techniques employed, which utilize copulas and their
properties, show that the bounds are pointwise best-possible when X and Y are symmetric about 0 and
yield expressions which can be evaluated explicitly when X and Y have a common distribution function
which is concave on (0,∞). Similar results are obtained for the radial error (X2+Y2)½. The important
case where X and Y are normally distributed is discussed, and here best-possible bounds on the circular
probable error are also obtained.