1 Robust Mean Estimation via Concentration Inequalities Suppose we observe a sequence of i.i.d. random variables X1, . . . , Xn. Their distribution is unknown, and has unknown mean u and known variance 02. In this question, we will investigate two different estimators for the mean u: the sample mean, and the socalled \"median of means\" estimator. In particular, we will analyze them in terms of how many samples n are required to estimate it to a given precision e and for a condence threshold 5. We'll start with the sample mean for parts (a) (c): in other words, we'll use X1, . . . , Xn to compute an estimate Sn 2 EzX for the mean n. We want to see what sample size it guarantees that lP(|[i u] 2 e) S 6. (d) (2 points) Fix a sample size no = . For each of the group means i, we define a binary random variable Zi: Zi = 1(15() - 1/ Z E). In other words, Z; is 1 if the corresponding group mean is close to the true mean / (within E), and 0 otherwise. Show that E[Zi]