I'm trying to find the probability density function (pdf) of the random variable U = X/Y where X and Y are both iid Cauchy random variables, the pdf of X is f(x) = a/(pi (a^2 + x^2)) and the pdf of Y is g(y) = a/(pi (a^2 + y^2)).
According to Wikipedia, the result is meant to be h(u) = a^2/(pi^2 (u^2 - a^4)) ln(u^2/a^4) but I keep getting 1/(pi^2 (u^2 - 1)) ln(u^2), which is independent of a ....
h(u) = int(-oo, +oo, |y| f(uy) g(y), dy)
= 2a^2/pi^2 int(0, +oo, y/((a^2 + u^2 y^2)(a^2 + y^2)), dy)
= 2/pi^2 int(0, +oo, t/((1 + u^2 t^2)(1 + t^2)), dt)
where t = y/a
= 1/(pi^2 (u^2 - 1)) ln(u^2).
Any comment on where I'm going wrong would be appreciated because I can't see my error (if there is one).
Thankyou in advance.
Itleung we appreciate your contributions to the community. I've been wondering, however, why you're responding to so many old threads? Quite a few of the threads you've replied to haven't had a response in ~2 years. It's fine to respond to old threads - it's just not typical behavior to focus on old threads.
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