exercises for 17 Feb to 23 Feb

Last week (10. Feb to 16. Feb) we talked about pi-systems and d-systems. We proved that a d-system that is also at pi-system is a sigma-algebra. Then we proved Dynkin's lemma, and the corollary saying that two finite measures that agree on a pi-system and agree and are equal when evaluated \Omega, agree on the sigma-algebra generated by that pi-system. On Thursday we proved a lemma about conditional expectation, then we derived the inversion formula, and proved that c.f.'s are unique. Finally, we derived an inversion expression for the density of random variables whose c.f. are integrable. 

Exercises: 2.17, 2.18, 2.20, 2.27, 2.28, 2.31, 2.38