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Statistics: Show that the sum of two independent Cauchy random variables is Cauchy. 
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#1
Feb1512, 12:40 PM

P: 37

Given the fact that X and Y are independent Cauchy random variables, I want to show that Z = X+Y is also a Cauchy random variable.
I am given that X and Y are independent and identically distributed (both Cauchy), with density function f(x) = 1/(∏(1+x2)) . I also use the fact the convolution integral for X and Y is ∫f(x)f(yx)dx . My book says to use the following hint: 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 : ) 


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