Proving Cauchy Density Function for Z = X+Y

[glacier302]

Homework Statement

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.

Homework Equations

Density function for X and Y is f(x)=1/(∏(1+x²)) .
Convolution integral = ∫f(x)f(y-x)dx .

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 : )

 

[vela]

Just expand out the righthand side and simplify.