  In probability theory, there is a tendency to treat one random variable with
a given distribution as being just as good as any other. By and large this is
fine because probability is (mostly) concerned with distributional properties
of random variables. However, every now and again we are forced to deal with
non-distributional properties. In this paper we investigate how different
random variables with the same distribution can be. Specifically, we construct
random variables that are all uniformly distributed on the unit interval, but
that nonetheless have strikingly different properties.
