Accept on faith that the characteristic function of the Cauchy distribution with density fX (x) = 1/(?(1+x2)) for ?? < x < ? is C(s) = E(e^isX) = e^?|s|. Suppose that X and Y are independent, and each has the Cauchy distribution. Find the characteristic function of Z = (X + Y )/2. What does this imply about the distribution of the mean of two Cauchy random variables?