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

  1. Feb 15, 2012 #1
    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(y-x)dx .

    My book says to use the following hint:

    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 15, 2012 #2

    mathman

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    Re: Statistics: Show that the sum of two independent Cauchy random variables is Cauch

    I can't see what to do with the hint. There is an alternate approach which might be easier if you have been exposed to characteristic functions (Fourier transform of densities).
    If you have, then get the characteristic function of Cauchy, square it and then get the inverse transform. This is the density that you want.
     
  4. Feb 16, 2012 #3

    Stephen Tashi

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    Re: Statistics: Show that the sum of two independent Cauchy random variables is Cauch

    I notice than in several statistics texts, this result is asserted or assigned as an exercise and the indicated method is expand the integrand in partial fractions. Perhaps the identity would be the result of that.
     
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