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.

Related Topics:
Paradox - Probability theory - Conditional probability - Density functions

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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) = rac{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 condition density for V given U.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:p_{V|U}(v|u) = rac{p_{V,U}(u,v)}{p_{U}(u)}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~