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

  1. Feb 15, 2012 #1
    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. jcsd
  3. Feb 17, 2012 #2

    vela

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

    Just expand out the righthand side and simplify.
     
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