Exercise 7 (Multiple Choice) For each of the following statements, mark correct ones with '[Y]' and incorrect statements with [N]' (1 point for each correct marking, we subtract 1 point for each wrong marking). a) [ ] Given i.i.d. samples X1, ..., Xn, S = n S(Xi - X)2 i=1 is an unbiased estimator for the standard deviation vVARIXi]. [ ] Maximum likelihood estimators are always unbiased. [ ] Given i.i.d. samples X1, ..., Xn, the sample mean X is an unbiased estimator for the population mean E[Xi]. b) You want to estimate the probability p that a biased coin shows tails. Your prior distribution is Beta(2, 2), and you see 10 heads in a row. Which of the following statements about the posterior P is true? Note that the density of Beta(r, q) is proportional to a"-1(1 - x)-1. [ ] P is a Beta distribution. [ ] P(p = 0) = 0. [ ] P(p = 1) =0. [ ] The MAP is 0. [ ] P(p P(p > 0.5). c) Let X and Y denote two RVs on the same probability space. Which of the following statements are true? [ ] X and Y cannot be independent. X and Y are always correlated. If X and Y are independent then X and Y are correlated. If X and Y are correlated then X and Y are independent. It is always possible to define a RV Z that is independent from X and Y. It is always possible to define a RV Z that is correlated to X and correlated to Y. [ ] X + Y cannot be independent of X