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
Figure 23.1 The support of (U, V) lies in the region −u < v < −u + 1.
For 0 < v < 1, the marginal density of V is
Using either probability density ...
