Problem 2 (5 points). For n N, let X1, ..., X, be independent identically distributed random...

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Problem 2 (5 points). For n N, let X1, ..., X, be independent identically distributed random variables on some discrete probability space (Q, A, P). Suppose E[X] < and define 2 Var (X). Now consider the empirical variance: == S(X) = - n 1 n Xi - x) 2 Show the following equations: i) Sh(X) = n(n-1) Eisiin (X - Xj). ii) E[S(X)] = . (Cf. unbiasedness of S2 (X) as an estimator of the variance.) Problem 2 (5 points). For n N, let X1, ..., X, be independent identically distributed random variables on some discrete probability space (Q, A, P). Suppose E[X] < and define 2 Var (X). Now consider the empirical variance: == S(X) = - n 1 n Xi - x) 2 Show the following equations: i) Sh(X) = n(n-1) Eisiin (X - Xj). ii) E[S(X)] = . (Cf. unbiasedness of S2 (X) as an estimator of the variance.)