1. The problem statement, all variables and given/known data
Let X and Y be independent random variables each having the Cauchy density function f(x)=1/(∏(1+x^{2})), and let Z = X+Y. Show that Z also has a Cauchy density function.
2. Relevant equations
Density function for X and Y is f(x)=1/(∏(1+x^{2})) .
Convolution integral = ∫f(x)f(yx)dx .
3. The attempt at a solution
My book gives the following hint, saying to "check it":
f(x)f(yx) = (f(x)+f(yx))/(∏(4+y^{2})) + 2/(∏y(4+y^{2}))(xf(x)+(yx)f(yx)) .
Using this hint, I'm able to solve the rest of the problem, but I can't figure out how to prove that this hint is true.
Any help would be much appreciated : )
