Density and Distribution Functions : Cauchy Density Function and Joint Distribution

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?

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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 …

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