1. Let X1, ..., Xn be an iid sample from a distribution F with density F' = f, and consider the KDE with a uniform kernel: n fm,k (I ) = _ nh i= 1 where K(t) = 1[-1/2,1/2] (t), and h is called the bandwidth, and 1A(t) denotes the indicator function of a set A, i.e. 1A(t) = 1 for r E A 10 for & # A' (Also note that K(t) is the density of the U([-}, }]) distribution.) Let Ph = F(I+ ?) - F(x -?). (a) Show that E(fn,k(2)) = Ph and Var(fn,k(x)) = - 1 nhi Ph(1 - Ph). HINT: Notice that if Yl, ...; Ym are iid, then ?_, 1A()?) has a binomial distri- bution. Why? (Think of Bernoulli trials...) Use this to find the distribution of Zi=1 1[-1/2,1/2] (1 2 ), and note that this equals nh fr, (I)