Borel's paradox

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

where p_X(x) is the appropriate marginal distribution.

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

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

A concrete example

A uniform distribution

We are given the joint probability density

Figure 1 shows the support of this distribution.

The marginal density of X is calculated to be

So the conditional density of Y given X is

which is uniform with respect to y.

Reparameterization

Now, we apply the following transformation:

Using the substitution rule, we obtain

Figure 2 shows the support of this distribution.

The marginal distribution is calculated to be

So the conditional density of V given U is

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

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

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

See also: Émile Borel

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