# 3. Consider a random sample of sample size n, {Xi}ni=1. We are interested in estimating the population mean of Xi, i.e, μ = E[X]. Let the sample average be Xn

3. Consider a random sample of sample size n, {Xi}ni=1. We are interested in estimating the population mean of Xi, i.e, μ = E[X]. Let the sample average be  Xn = (1/n) ∑n i=1 Xi. Let σ2 = Var(X) be the variance of Xi. (a) Prove that V ar( ̄Xn) = σ2 n . will happen to the variance of ̄Xn as n →∞? (b) Prove that Var(√n( ̄Xn −μ)) = σ2. (c) Discuss what will happen to the variance of a random variable Wn = n3/4( ̄Xn −μ) as n →∞. will happen to the distribution of Wn = n3/4( ̄Xn−μ) as n →∞? Consider a random sample of sample size n, {Xi}" ]. We are interested in estimat
ing the population mean of Xi, i.e, u = E[X]. Let the sample average be Xn
=
(1/n) Et Xi. Let o2 = Var(X) be the variance of Xi.
(a) Prove that Var(Xn) = . will happen to the variance of Xn as n – co?
(b) Prove that Var(Vn(Xn -M)) = 02.
(c) Discuss what will happen to the variance of a random variable Wn = n3/4(X, -/)
as n -> co. will happen to the distribution of Wn = n3/4(Xn-u) as n -> co?

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