Question

2. Biased and unbiased estimation for variance of Bernoulli variables A Bookmark this page 2 points possible (graded) Let X1, X, bed. Bernoull random variables, with unknown parameter PE (0,1). The aim of this exercise is to estimate the common variance of the X First, recall what Var (X) is for Bernoulli random variables. Var (X) - Let X, be the sample average of the Xi. X. - 3x Interested in finding an estimator for Var(X), and propose to use V-X.(1-X.). Check the correct statement that applies to V: O V is not consistent because Var (1) is not linear in P O V is consistent because of the Law of large Numbers O is consistent because of the Central Limit Theorem STANDARD NOTATION You have used of 2 attempts

Answer 1

var (X;) = P (1-P) = p - p2 ] - û = xn (1-in) û is not consistent because var (xi) is not linear in P - Now ECO) - ES X (1-10) 3 .. - E [ XD - 2] : - E(XD) - E(803). - <?(353) - [var (8) + [Ex.»" - yax(n) - E(X) - [ECE] : I E(X) - [var (sx;) + FE (45%)]? ECO = hp - [1 hP9 + ( np.) ² ECO) = P - Pq - p2 - NOLO ECO) - war (xi) = P - Pq -P²-(Pq). - P(1-P)-Pflent = pq - Pq (1+1)

P9 (I-18h) LE(W)-var(x;) - Pq. n. from E (O) = = o P-PL - p² P(1-P) - pq = P9 - pq TE(O) = pq (1-1)