Stat 5101 Lecture Notes - School of Statistics

6.1. UNIVARIATE THEORY 175Problems6-1. Derive (6.9) from (6.8) using the change of variable theorem.6-2. Suppose that S 1 , S 2 , ... is any sequence of random variables such thatPS n −→ σ, and X1 , X 2 , ... are independent and identically distributed withmean µ and variance σ 2 and σ>0. Show thatX n − µS n / √ nD−→ N (0, 1),as n →∞,where, as usual,X n = 1 n6-3. Suppose X 1 , X 2 , ... are i. i. d. with common probability measure P , anddefine Y n = I A (X n ) for some event A, that is,{1, X n ∈ AY n =0, X n /∈ AShow that Y nP−→ P (A).6-4. Suppose the sequences X 1 , X 2 , ... and Y 1 , Y 2 , ... are defined as in Problem6-3, and write P (A) =p. Show that√ n(Y n − p)n∑i=1X i−→ DN ( 0,p(1 − p) )and also show that√Y n − p D−→ N (0, 1)Y n (1 − Y n )/nHint: What is the distribution of ∑ i Y i? Also use Problem 6-2.