O'Reilly logo

Classic Problems of Probability by Prakash Gorroochurn

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

Problem 23

Borel's Paradox and Kolmogorov's Axioms (1909, 1933)

Problem. Suppose the random variables X and Y have the joint density function

img

Obtain the conditional density of X, given that Y = X.

Solution. Let img and img. Then we aim to find img. We have

img

where J is the Jacobian

img

Therefore, img for img (see Fig. 23.1). The marginal density of V for −1 < v < 0 is

equation

Figure 23.1 The support of (U, V) lies in the region −u < v < −u + 1.

img

For 0 < v < 1, the marginal density of V is

Using either probability density ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required