I only need the fourth question; sorry for inconvenient cropping but I was suggest to post the whole problem set to clarify the fourth question

I only

3 Problem Set: Inequalities... 1. Let (P1, ..., Pn) and (91, ..., qn) be two probabilistic distributions, e.g Pi, qi > 0 for all i; isisn Pi = Eisisngi = 1. Define KLD(P||q) = pi(log(Pi) log(qi)). 1 (Eisisn\pi 9:])? 4. Prove that KLD(p||) > 1(Eisisn |Pi - 4:]). 5. (Big Bonus Question! The last inequality is pretty famous What is its name? 3 Problem Set: Inequalities... 1. Let (P1, ..., Pn) and (91, ..., qn) be two probabilistic distributions, e.g Pi, qi > 0 for all i; isisn Pi = Eisisngi = 1. Define KLD(P||q) = pi(log(Pi) log(qi)). 1 (Eisisn\pi 9:])? 4. Prove that KLD(p||) > 1(Eisisn |Pi - 4:]). 5. (Big Bonus Question! The last inequality is pretty famous What is its name