# Statistics: Show that sum of two independent Cauchy random variables is also Cauchy

1. Feb 15, 2012

### glacier302

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 : )

2. Feb 17, 2012

### vela

Staff Emeritus
Re: Statistics: Show that sum of two independent Cauchy random variables is also Cauc

Just expand out the righthand side and simplify.