Problem 23

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 ...