4. (2 points) Here is an alternative way to estimate F2. Let h1,..., hd be hash functions from [1, N] to [1, N), randomly chosen from a 4-wise independent family. Let c[r][i] = Ej:h+6)=i fj. Suppose that w is an even integer. For each re [1, d), let w/2 er = (c[r][21] c[+][2i 1]). i=1 The algorithm will output the median of 1, ..., ed as the estimate of F2. Show the following: E [21] = F2. 4. (2 points) Here is an alternative way to estimate F2. Let h1,..., hd be hash functions from [1, N] to [1, N), randomly chosen from a 4-wise independent family. Let c[r][i] = Ej:h+6)=i fj. Suppose that w is an even integer. For each re [1, d), let w/2 er = (c[r][21] c[+][2i 1]). i=1 The algorithm will output the median of 1, ..., ed as the estimate of F2. Show the following: E [21] = F2