Statistics: Show that sum of two independent Cauchy random variables is also Cauchy
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+x2)), 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+x2)) .
Convolution integral = ∫f(x)f(y-x)dx .
3. The attempt at a solution
My book gives the following hint, saying to "check it":
f(x)f(y-x) = (f(x)+f(y-x))/(∏(4+y2)) + 2/(∏y(4+y2))(xf(x)+(y-x)f(y-x)) .
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 : )