4.17. X has the U(?pi/2, pi/2) distribution, and Y = tan(X). Show that V has density l/(pi(1 + y2)) for ?oo <y <oo . (This is the Cauchy density function.) What can be said about the meanand varianceof Y? How could you simulate values from this distribution, given a supply of U(O, 1) values?

4.21. Let X and Y have joint density 2 exp(?x ? y) over 0 < x <y < oo. Find their marginal densities; the density of X, given Y = 4; and the density of Y, given X = 4. Show that X and Y are not independent.

Find the joint density of U = X + Y and V = X/Y. Are U and V independent?

Please see the attached file for the complete solution.

Thanks for using BrainMass.

Solution to 4.17.

In order to find the density function of Y, we need to find its distribution function first. Since X follows uniformly over , we know that its density function is as follows.

Now , so

So, the density function of Y is given by

Note:

Then by definition of mean, we can compute the mean of Y as follows.

This is an improper …

The Cauchy Density Function and Joint Distributions are investigated. The solution is detailed and well presented. The response received a rating of “5” from the student who originally posted the question.