Borel's paradox (sometimes known as the Borel-Kolmogorov paradox) is a paradox of probability theory relating to conditional probability density functions. Robert Boyles self-flowing flask fills itself in this diagram, but perpetual motion machines do not exist. ... Probability theory is the mathematical study of probability. ... This article defines some terms which characterize probability distributions of two or more variables. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

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, A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...

where p_X(x) is the appropriate marginal distribution. Given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y, typically calculated by summing or integrating the joint probability distribution over Y. For discrete random variables, the marginal probability mass function can be written as...

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. In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ...

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

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.

Reparametrization

Now, we apply the following transformation:

Using the substitution rule, we obtain

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