If f is the pdf of the standard normal distribution show

Question

Let X₁, ..., X_n be i.i.d. with a Lebesgue p.d.f. f(x - µ), where f is known and µ is unknown.

(a) If f is the p.d.f. of the standard normal distribution, show that the confidence interval [X - c₁, X + c₁] is better than [X₁ - c₂, X₁ +c₂] in terms of their lengths, where ci's are chosen so that these confidence intervals have confidence coefficient 1 - α.

(b) If f is the p.d.f. of the Cauchy distribution C(0, 1), show that the two confidence intervals in (a) have the same length.