Suppose that a random variable has a uniform distribution on a unit sphere. What is its conditional distribution on a great circle? Because of the symmetry of the sphere, one might expect that the distribution is uniform and independent of the choice of coordinates. However, two analyses give contradictory results. First, note that choosing a point uniformly on the sphere is equivalent to choosing the longitudeλ uniformly from and choosing the latitudeφ from with density. Then we can look at two different great circles:
If the coordinates are chosen so that the great circle is an equator , the conditional density for a longitude λ defined on the interval is
One distribution is uniform on the circle, the other is not. Yet both seem to be referring to the same great circle in different coordinate systems.
Explanation and implications
In case above, the conditional probability that the longitude λ lies in a set E given that φ = 0 can be written P. Elementary probability theory suggests this can be computed as P/P, but that expression is not well-defined since P = 0. Measure theory provides a way to define a conditional probability, using the family of events Rab = which are horizontal rings consisting of all points with latitude between a and b. The resolution of the paradox is to notice that in case, P is defined using the events Lab =, which are lunes, consisting of all points whose longitude varies between a and b. So although P and P each provide a probability distribution on a great circle, one of them is defined using rings, and the other using lunes. Thus it is not surprising after all that P and P have different distributions.
Mathematical explication
To understand the problem we need to recognize that a distribution on a continuous random variable is described by a density f only with respect to some measure μ. Both are important for the full description of the probability distribution. Or, equivalently, we need to fully define the space on which we want to define f. Let Φ and Λ denote two random variables taking values in Ω1 = respectively Ω2 = . An event gives a point on the sphere S with radius r. We define the coordinate transform for which we obtain the volume element Furthermore, if either φ or λ is fixed, we get the volume elements Let denote the joint measure on, which has a density with respect to and let If we assume that the density is uniform, then Hence, has a uniform density with respect to but not with respect to the Lebesgue measure. On the other hand, has a uniform density with respect to and the Lebesgue measure.
