16 pts) PROBLEM 21. Let f:X →Y be a function, let Xi, X2SX andlet Yi, ½SY. ) Write down the definitions of f(Xi) and f (Y 。ín½) = f-'(%) nf-106). (ii) Prove that (ii) Prove that f(XinX)(xnf(...
f Squares and Properties of Estimators o. Let xi yi denote two series ofn numbers xi: i-1,2...), tyi: i 1,2...n) Assume that xi s drawn from a distribution that is NOHm σ) Show that the sample mean i ΣΙ-1 χί has a variance of σ/n carefully stating any required assunmptions at each step. Is the sample mean an unbiased estimator of u,? 1. ii. The following results are useful when working with linear regressions. Show that: 2 iii. Show that:...
Problem II i) Theorem 2.9 in the course text states that a function f: X → Y is continuous if and only if f(A) C (A) for all A CX. Formulate and prove an analogous statement for A ii) Show that J: X → Y is continuous if and only if f: X → f(X) is continuous Here f(p) = f(p) for all p E X and f(X) c Y ls equipped with the subspace topology Problem II i) Theorem...