Problem 6 Let {$1, $2, . . . ,rrn, . . .} be a sequence of independent random variables with identical means lE[:ri] = m and variances var(a:i) = (:2. Dene the sample mean and sample mean-square of the rst N of the mi's as sample mean: ZHIH 2N! || 2' H sample meansquare: =i i 1. Find the mean and variance of the sample mean. Show that . _ 2 = ngnmEKmN m) 1 0 and use this result to deduce that Alim Pr[|mN m|_ > e] 0 for any 6 > O. . Suppose that {$1,1'2,.. . ,3:m. . .} have identical means, but are not necessarily independent. ls the mean of the sample mean the same as part 1? Is the variance of the sample mean the same as part 1? 3. Suppose the mi are independent, zeromean Gaussian random variables. Find the mean and variance of the sample meansquare. Show that $130 1E[(S?v - 02V] = 0