Let {Xi} n i=1 be a random sample from a population with E [X] = 2, E [X2 ] = 6, E [X3 ] = 1 and E [X4 ] = 37. (a) Let X¯ n = 1 n Xn i=1 Xi . Compute the probability limit of 3 X¯ n + 1 as n → [infinity]. Please state all the results you have used to obtain your answer, explaining the reasons of why you can use them (generic answers are not accepted).
(b) Let g¯n (X) = 1 n Xn i=1 X 3 i . Compute the probability limit of 2¯gn (X) as n → [infinity]. Please state all the results you have used, explaining the reasons of why you can use them (generic answers are not accepted).
(c) What distribution does √ n 5X¯ n − 10 2 converge to? Please state all the results you have used, and explain the reasons of why you can use them (generic answers are not accepted). The mean and variance of the limit distribution should be presented as numbers, and not in generic terms.