Show that the sample means for the two sets are close. Specically, show: are) are) 5 2 (2) This is referred to aspn having sensitivity nil. (b) (1 point) For any xedS, explain why pg,\" (5) is distributed according to a Laplace distribution. What are the corresponding parameters? (c) (2 points) First, we will show that the above estimator is still fairly accurate. Show that with probability 099 (over the sampling of the noise), for everys, we have pasgame g 2 You may find it especially useful to apply a concentration inequality we learned about in class. ((1) In this part, we will see that the mechanism isdifferentially private. Let us recall the denition of differential privacy in this context. An estimatog is Edifferentially private if for all setsA C R, we have Pr[g(51) E A] g exp(e) ' Pr 9(52) 6 A where S1, 82 are two data sets that differ only in one item. (i) {3 points) Let Y1 N Lap (#1, b) and Y2 N Lap ((12, b). Show that -Pr [Y2 E A] . This hints at why the Laplace distribution is particularly well suited for differential privacy. Hint: Find a bound on the likelihood ratio, and. relate that to the inequality above (ii) (2 points) Using Equation (2) and earlier parts of the question, show that the esti- mator lie,\" is Edifferentially private. (iii) (1 points) Put these steps together show than}; is a 6-DP estimator for p with error 1 1 3315511 S0 \"+' ne with probability 098 over the randomness of the sample and the mechanism. (e) (1 point) Now, suppose that instead of Bernoulli, the individual sample-3Y1: were real- valued random variables taking values in [05]. Which part(s) of the analysis above (if any) would change? You don't need to redo the analysis