1- Let X1, X2, ... be an iid sequence of rv's each with finite mean Ī¼, finite variance ĻĀ² > 0 and mgf M(t) defined for |t| < h with h > 0.
(a) Define the sample variance SāĀ² and show that the sequence {SāĀ²} provides a consistent estimator for ĻĀ².
(b) Show that for all t with 0 < |t| < h, M(t) > e^tĪ¼.
(c) Show that for |t| < h/2 the sequence {1/n āįµ¢āāāæ e^tXįµ¢} converges in probability and the sequence {[nM(t) - āįµ¢āāāæ e^tXįµ¢] / [ā(n(M(2t) - M(t)Ā²))]} converges in distribution. In each case, describe the limit.