Question

(3) Let XXnX1,X2,⋯,Xn be iidiid random variables with Cauchy(0,1)Cauchy(0,1) distribution. That is, the density of X1 is 1/(π(1+x2)) for x∈ℜ. Prove that (X1+X2+⋯+Xn)/n is again distributed as Cauchy(0,1).

The following ``answers'' have been proposed. Please read the choices very carefully and pick the most complete and accurate choice.

(a) By the last exercise, the characteristic function of X1, is e−|t|e−|t|. Therefore by the fact that the Xi are iid, the characteristic function of their average is the product of n terms each of which is e−|t|/ne−|t|/n, which finishes the result.
(b) By the last exercise, the characteristic function of X1, is e−|t|e−|t|. Therefore by the fact that the XiXi are iid, the characteristic function of their average is average of the individual characteristics functions, namely the average of n terms, each one of them being e^−|t|. The resulting average is therefore e^−|t|
(c) By the last exercise, the characteristic function of X1, is e−|t|e−|t|. Therefore by the fact that the XiXi are iid, the characteristic function of their average converges to e^−|t| by the usual central limit theorem for iid random variables.
(d) Actually the statement of the problem is false, because if this were to be true, it will invalidate the central limit theorem as n gets large.
(e) None of the above

The correct answer is

Answer 1