univariate
06-18-2010, 09:54 PM
Hello.
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 ....
My work:
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.
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 ....
My work:
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.