Borel's Paradox and Kolmogorov's Axioms (1909, 1933)
Problem. Suppose the random variables X and Y have the joint density function
Obtain the conditional density of X, given that Y = X.
Solution. Let and . Then we aim to find . We have
where J is the Jacobian
Therefore, for (see Fig. 23.1). The marginal density of V for −1 < v < 0 is
For 0 < v < 1, the marginal density of V is
Using either probability density ...