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.