Borel's paradox

Borel's paradox (sometimes known as the Borel-Kolmogorov paradox) is a paradox of probability theory relating to conditional probability density functions. The paradox lies in fact that, contrary to intuition, conditional probability density functions are not invariant under coordinate transformations.

Suppose we have two random variables, X and Y, with joint probability density pX,Y(x,y). We can form the conditional density for Y given X,


 * $$p_{Y|X}(y|x) = \frac{p_{X,Y}(x,y)}{p_{X}(x)}$$

where pX(x) is the appropriate marginal distribution.

Using the substitution rule, we can reparametrize the joint distribution with the functions U= f(X,Y), V = g(X,Y), and can then form the conditional density for V given U.


 * $$p_{V|U}(v|u) = \frac{p_{U,V}(u,v)}{p_{U}(u)}$$

Given a particular condition on X and the equivalent condition on U, intuition suggests that the conditional densities pY(y|x) and pV(v|u) should also be equivalent. This is not the case in general.

A uniform distribution
We are given the joint probability density


 * $$p_{X,Y}(x,y) =\left\{\begin{matrix} 1, & 0 < y < 1, \quad -y < x < 1 - y \\ 0, & \mbox{otherwise} \end{matrix}\right. $$

The marginal density of X is calculated to be


 * $$p_X(x) =\left\{\begin{matrix} 1+x, & -1 < x \le 0 \\ 1 - x, & 0 < x < 1 \\ 0, & \mbox{otherwise}\end{matrix}\right. $$

So the conditional density of Y given X is


 * $$p_{Y|X}(y|x) =\left\{\begin{matrix} \frac{1}{1+x}, & -1 < x \le 0, \quad -x < y < 1 \\ \\ \frac{1}{1-x}, & 0 < x < 1, \quad 0 < y < 1 - x \\ \\ 0, & \mbox{otherwise}\end{matrix}\right. $$

which is uniform with respect to y.

Reparametrization
Now, we apply the following transformation:


 * $$U = \frac{X}{Y} + 1 \qquad \qquad V = Y.$$

Using the substitution rule, we obtain


 * $$p_{U,V}(u,v) =\left\{\begin{matrix} v, & 0 < v < 1, \quad 0 < u \cdot v < 1 \\ 0, & \mbox{otherwise} \end{matrix}\right. $$

The marginal distribution is calculated to be


 * $$p_U(u) =\left\{\begin{matrix} \frac{1}{2}, & 0 < u \le 1 \\ \\ \frac {1}{2u^2}, & 1 < u < +\infty \\ \\ 0, & \mbox{otherwise}\end{matrix}\right. $$

So the conditional density of V given U is


 * $$p_{V|U}(v|u) =\left\{\begin{matrix} 2v, & 0 < u \le 1, \quad 0 < v < 1 \\ 2u^2v, & 1 < u < +\infty, \quad 0 < v < \frac{1}{u} \\ 0, & \mbox{otherwise}\end{matrix}\right. $$

which is not uniform with respect to v.

The unintuitive result
Now we pick a particular condition to demonstrate Borel's paradox. The conditional density of Y given X = 0 is


 * $$p_{Y|X}(y|x=0) = \left\{\begin{matrix} 1, & 0 < y < 1 \\ 0, & \mbox{otherwise}\end{matrix}\right. $$

The equivalent condition in the u-v coordinate system is U = 1, and the conditional density of V given U = 1 is


 * $$p_{V|U}(v|u=1) = \left\{\begin{matrix} 2v, & 0 < v < 1 \\ 0, & \mbox{otherwise}\end{matrix}\right. $$

Paradoxically, V = Y and X = 0 is equivalent to U = 1, but


 * $$p_{Y|X}(y|x = 0) \ne p_{V|U}(v|u = 1).\;$$