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    Density and Distribution Functions : Cauchy Density Function and Joint Distribution

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    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 mean and variance of 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


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

    This is an improper ...

    Solution Summary

    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.